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</div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Appearance</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" 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style="display:none">Property of topological spaces</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Neighborhood_illust1.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/79/Neighborhood_illust1.svg/220px-Neighborhood_illust1.svg.png" decoding="async" width="220" height="213" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/79/Neighborhood_illust1.svg/330px-Neighborhood_illust1.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/79/Neighborhood_illust1.svg/440px-Neighborhood_illust1.svg.png 2x" data-file-width="518" data-file-height="502" /></a><figcaption>In this topological space, <i>V</i> is a neighbourhood of <i>p</i> and it contains a connected open set (the dark green disk) that contains <i>p</i>.</figcaption></figure> <p>In <a href="/wiki/Topology" title="Topology">topology</a> and other branches of <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, a <a href="/wiki/Topological_space" title="Topological space">topological space</a> <i>X</i> is <b>locally connected</b> if every point admits a <a href="/wiki/Neighbourhood_basis" class="mw-redirect" title="Neighbourhood basis">neighbourhood basis</a> consisting of <a href="/wiki/Open_set" title="Open set">open</a> <a href="/wiki/Connected_set" class="mw-redirect" title="Connected set">connected</a> sets. </p><p>As a stronger notion, the space <i>X</i> is <b>locally path connected</b> if every point admits a neighbourhood basis consisting of open <a href="/wiki/Path_connected" class="mw-redirect" title="Path connected">path connected</a> sets. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Background">Background</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Locally_connected_space&action=edit&section=1" title="Edit section: Background"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Throughout the history of topology, <a href="/wiki/Connected_space" title="Connected space">connectedness</a> and <a href="/wiki/Compact_space" title="Compact space">compactness</a> have been two of the most widely studied topological properties. Indeed, the study of these properties even among subsets of <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a>, and the recognition of their independence from the particular form of the <a href="/wiki/Euclidean_metric" class="mw-redirect" title="Euclidean metric">Euclidean metric</a>, played a large role in clarifying the notion of a topological property and thus a topological space. However, whereas the structure of <i>compact</i> subsets of Euclidean space was understood quite early on via the <a href="/wiki/Heine%E2%80%93Borel_theorem" title="Heine–Borel theorem">Heine–Borel theorem</a>, <i>connected</i> subsets of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span> (for <i>n</i> > 1) proved to be much more complicated. Indeed, while any compact <a href="/wiki/Hausdorff_space" title="Hausdorff space">Hausdorff space</a> is <a href="/wiki/Locally_compact" class="mw-redirect" title="Locally compact">locally compact</a>, a connected space—and even a connected subset of the Euclidean plane—need not be locally connected (see below). </p><p>This led to a rich vein of research in the first half of the twentieth century, in which topologists studied the implications between increasingly subtle and complex variations on the notion of a locally connected space. As an example, the notion of local connectedness im kleinen at a point and its relation to local connectedness will be considered later on in the article. </p><p>In the latter part of the twentieth century, research trends shifted to more intense study of spaces like <a href="/wiki/Manifold" title="Manifold">manifolds</a>, which are locally well understood (being <a href="/wiki/Locally_homeomorphic" class="mw-redirect" title="Locally homeomorphic">locally homeomorphic</a> to Euclidean space) but have complicated global behavior. By this it is meant that although the basic <a href="/wiki/Point-set_topology" class="mw-redirect" title="Point-set topology">point-set topology</a> of manifolds is relatively simple (as manifolds are essentially <a href="/wiki/Metrizable" class="mw-redirect" title="Metrizable">metrizable</a> according to most definitions of the concept), their <a href="/wiki/Algebraic_topology" title="Algebraic topology">algebraic topology</a> is far more complex. From this modern perspective, the stronger property of local path connectedness turns out to be more important: for instance, in order for a space to admit a <a href="/wiki/Universal_cover" class="mw-redirect" title="Universal cover">universal cover</a> it must be connected and locally path connected. </p><p>A space is locally connected if and only if for every open set <span class="texhtml">U</span>, the connected components of <span class="texhtml">U</span> (in the <a href="/wiki/Subspace_topology" title="Subspace topology">subspace topology</a>) are open. It follows, for instance, that a continuous function from a locally connected space to a <a href="/wiki/Totally_disconnected" class="mw-redirect" title="Totally disconnected">totally disconnected</a> space must be locally constant. In fact the openness of components is so natural that one must be sure to keep in mind that it is not true in general: for instance <a href="/wiki/Cantor_space" title="Cantor space">Cantor space</a> is totally disconnected but not <a href="/wiki/Discrete_space" title="Discrete space">discrete</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Definitions">Definitions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Locally_connected_space&action=edit&section=2" title="Edit section: Definitions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> be a topological space, and let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> be a point of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ba76c5a460c4a0bb1639a193bc1830f0a773e03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}"></span> </p><p>A space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> is called <b>locally connected at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span></b><sup id="cite_ref-FOOTNOTEMunkres2000161_1-0" class="reference"><a href="#cite_note-FOOTNOTEMunkres2000161-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> if every <a href="/wiki/Neighborhood_(mathematics)" class="mw-redirect" title="Neighborhood (mathematics)">neighborhood</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> contains a <a href="/wiki/Connected_(topology)" class="mw-redirect" title="Connected (topology)">connected</a> <i>open</i> neighborhood of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>, that is, if the point <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> has a <a href="/wiki/Neighborhood_base" class="mw-redirect" title="Neighborhood base">neighborhood base</a> consisting of connected open sets. A <b>locally connected space</b><sup id="cite_ref-FOOTNOTEWillard2004199Definition_27.7_2-0" class="reference"><a href="#cite_note-FOOTNOTEWillard2004199Definition_27.7-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTEMunkres2000161_1-1" class="reference"><a href="#cite_note-FOOTNOTEMunkres2000161-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> is a space that is locally connected at each of its points. </p><p>Local connectedness does not imply connectedness (consider two disjoint open intervals in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span> for example); and connectedness does not imply local connectedness (see the <a href="/wiki/Topologist%27s_sine_curve" title="Topologist's sine curve">topologist's sine curve</a>). </p><p><span class="anchor" id="locally_path_connected"></span>A space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> is called <b>locally path connected at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span></b><sup id="cite_ref-FOOTNOTEMunkres2000161_1-2" class="reference"><a href="#cite_note-FOOTNOTEMunkres2000161-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> if every neighborhood of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> contains a <a href="/wiki/Path_connected" class="mw-redirect" title="Path connected">path connected</a> <i>open</i> neighborhood of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>, that is, if the point <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> has a neighborhood base consisting of path connected open sets. A <b>locally path connected space</b><sup id="cite_ref-FOOTNOTEWillard2004199Definition_27.4_3-0" class="reference"><a href="#cite_note-FOOTNOTEWillard2004199Definition_27.4-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTEMunkres2000161_1-3" class="reference"><a href="#cite_note-FOOTNOTEMunkres2000161-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> is a space that is locally path connected at each of its points. </p><p>Locally path connected spaces are locally connected. The converse does not hold (see the <a href="/wiki/Lexicographic_order_topology_on_the_unit_square" title="Lexicographic order topology on the unit square">lexicographic order topology on the unit square</a>). </p> <div class="mw-heading mw-heading3"><h3 id="Connectedness_im_kleinen">Connectedness im kleinen</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Locally_connected_space&action=edit&section=3" title="Edit section: Connectedness im kleinen"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> is called <b>connected im kleinen at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span></b><sup id="cite_ref-FOOTNOTEWillard2004201Definition_27.14_4-0" class="reference"><a href="#cite_note-FOOTNOTEWillard2004201Definition_27.14-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-BBS_5-0" class="reference"><a href="#cite_note-BBS-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> or <b>weakly locally connected at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span></b><sup id="cite_ref-FOOTNOTEMunkres2000162Exercise_6_6-0" class="reference"><a href="#cite_note-FOOTNOTEMunkres2000162Exercise_6-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> if every neighborhood of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> contains a connected (not necessarily open) neighborhood of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>, that is, if the point <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> has a neighborhood base consisting of connected sets. A space is called <b>weakly locally connected</b> if it is weakly locally connected at each of its points; as indicated below, this concept is in fact the same as being locally connected. </p><p>A space that is locally connected at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> is connected im kleinen at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d07e9f568a88785ae48006ac3c4b951020f1699a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.977ex; height:1.676ex;" alt="{\displaystyle x.}"></span> The converse does not hold, as shown for example by a certain infinite union of decreasing <a href="/wiki/Broom_space" class="mw-redirect" title="Broom space">broom spaces</a>, that is connected im kleinen at a particular point, but not locally connected at that point.<sup id="cite_ref-FOOTNOTESteenSeebach1995139Example_119.4_7-0" class="reference"><a href="#cite_note-FOOTNOTESteenSeebach1995139Example_119.4-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTEMunkres2000162Exercise_7_8-0" class="reference"><a href="#cite_note-FOOTNOTEMunkres2000162Exercise_7-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> However, if a space is connected im kleinen at each of its points, it is locally connected.<sup id="cite_ref-FOOTNOTEWillard2004201Theorem_27.16_10-0" class="reference"><a href="#cite_note-FOOTNOTEWillard2004201Theorem_27.16-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p><p>A space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> is said to be <b>path connected im kleinen at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span></b><sup id="cite_ref-BBS_5-1" class="reference"><a href="#cite_note-BBS-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> if every neighborhood of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> contains a path connected (not necessarily open) neighborhood of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>, that is, if the point <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> has a neighborhood base consisting of path connected sets. </p><p>A space that is locally path connected at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> is path connected im kleinen at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d07e9f568a88785ae48006ac3c4b951020f1699a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.977ex; height:1.676ex;" alt="{\displaystyle x.}"></span> The converse does not hold, as shown by the same infinite union of decreasing broom spaces as above. However, if a space is path connected im kleinen at each of its points, it is locally path connected.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup class="noprint Inline-Template noprint noexcerpt Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:NOTRS" class="mw-redirect" title="Wikipedia:NOTRS"><span title="This claim needs references to better sources. (December 2024)">better source needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading2"><h2 id="First_examples">First examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Locally_connected_space&action=edit&section=4" title="Edit section: First examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ol><li>For any positive integer <i>n</i>, the Euclidean space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span> is locally path connected, thus locally connected; it is also connected.</li> <li>More generally, every <a href="/wiki/Locally_convex_topological_vector_space" title="Locally convex topological vector space">locally convex topological vector space</a> is locally connected, since each point has a local base of <a href="/wiki/Convex_set" title="Convex set">convex</a> (and hence connected) neighborhoods.</li> <li>The subspace <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=[0,1]\cup [2,3]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo>∪</mo> <mo stretchy="false">[</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=[0,1]\cup [2,3]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26ef5909673801c27872765f3a55ce1c6f09e589" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.485ex; height:2.843ex;" alt="{\displaystyle S=[0,1]\cup [2,3]}"></span> of the real line <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec2e282911e406fc800fb1095093667d66f18c7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{1}}"></span> is locally path connected but not connected.</li> <li>The <a href="/wiki/Topologist%27s_sine_curve" title="Topologist's sine curve">topologist's sine curve</a> is a subspace of the Euclidean plane that is connected, but not locally connected.<sup id="cite_ref-FOOTNOTESteenSeebach1995137–138_12-0" class="reference"><a href="#cite_note-FOOTNOTESteenSeebach1995137–138-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup></li> <li>The space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }"></span> of <a href="/wiki/Rational_numbers" class="mw-redirect" title="Rational numbers">rational numbers</a> endowed with the standard Euclidean topology, is neither connected nor locally connected.</li> <li>The <a href="/wiki/Comb_space" title="Comb space">comb space</a> is path connected but not locally path connected, and not even locally connected.</li> <li>A countably infinite set endowed with the <a href="/wiki/Cofinite_topology" class="mw-redirect" title="Cofinite topology">cofinite topology</a> is locally connected (indeed, <a href="/wiki/Hyperconnected" class="mw-redirect" title="Hyperconnected">hyperconnected</a>) but not locally path connected.<sup id="cite_ref-FOOTNOTESteenSeebach199549–50_13-0" class="reference"><a href="#cite_note-FOOTNOTESteenSeebach199549–50-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup></li> <li>The <a href="/wiki/Lexicographic_order_topology_on_the_unit_square" title="Lexicographic order topology on the unit square">lexicographic order topology on the unit square</a> is connected and locally connected, but not path connected, nor locally path connected.<sup id="cite_ref-FOOTNOTESteenSeebach199573Example_48_14-0" class="reference"><a href="#cite_note-FOOTNOTESteenSeebach199573Example_48-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup></li> <li>The <a href="/wiki/Kirch_space" class="mw-redirect" title="Kirch space">Kirch space</a> is connected and locally connected, but not path connected, and not path connected im kleinen at any point. It is in fact <a href="/wiki/Totally_path_disconnected" class="mw-redirect" title="Totally path disconnected">totally path disconnected</a>.</li></ol> <p>A <a href="/wiki/First-countable" class="mw-redirect" title="First-countable">first-countable</a> <a href="/wiki/Hausdorff_space" title="Hausdorff space">Hausdorff space</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,\tau )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,\tau )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dede4b8004c6222d11e9b1a9802dc0496ad7e1fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.025ex; height:2.843ex;" alt="{\displaystyle (X,\tau )}"></span> is locally path-connected if and only if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }"></span> is equal to the <a href="/wiki/Final_topology" title="Final topology">final topology</a> on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> induced by the set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C([0,1];X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo stretchy="false">(</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo>;</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C([0,1];X)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/366864ca3d1510fa383601d72d0f0c73dc3f7e2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.242ex; height:2.843ex;" alt="{\displaystyle C([0,1];X)}"></span> of all continuous paths <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,1]\to (X,\tau ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,1]\to (X,\tau ).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c98675e3145644ae285fe1f33014c6e964a596bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.939ex; height:2.843ex;" alt="{\displaystyle [0,1]\to (X,\tau ).}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Locally_connected_space&action=edit&section=5" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1110004140">.mw-parser-output .math_theorem{margin:1em 2em;padding:0.5em 1em 0.4em;border:1px solid #aaa;overflow:hidden}@media(max-width:500px){.mw-parser-output .math_theorem{margin:1em 0em;padding:0.5em 0.5em 0.4em}}</style><div class="math_theorem" style=""> <p><strong class="theorem-name">Theorem</strong><span class="theoreme-tiret"> — </span>A space is locally connected if and only if it is weakly locally connected.<sup id="cite_ref-FOOTNOTEWillard2004201Theorem_27.16_10-1" class="reference"><a href="#cite_note-FOOTNOTEWillard2004201Theorem_27.16-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p> </div> <style data-mw-deduplicate="TemplateStyles:r1256386598">.mw-parser-output .cot-header-mainspace{background:#F0F2F5;color:inherit}.mw-parser-output .cot-header-other{background:#CCFFCC;color:inherit}@media screen{html.skin-theme-clientpref-night .mw-parser-output .cot-header-mainspace{background:#14181F;color:inherit}html.skin-theme-clientpref-night .mw-parser-output .cot-header-other{background:#003500;color:inherit}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cot-header-mainspace{background:#14181F;color:inherit}html.skin-theme-clientpref-os .mw-parser-output .cot-header-other{background:#003500;color:inherit}}</style> <div style="margin-left:0"> <table class="mw-collapsible mw-archivedtalk mw-collapsed" style="color:inherit; background: transparent; text-align: left; border: 1px solid Silver; margin: 0.2em auto auto; width:100%; clear: both; padding: 1px;"> <tbody><tr> <th class="cot-header-mainspace" style="; font-size:87%; padding:0.2em 0.3em; text-align:left;"><div style="font-size:115%;">Proof</div> </th></tr> <tr> <td style="color:inherit; border: solid 1px Silver; padding: 0.6em; background: var(--background-color-base, #fff);"> <p>For the non-trivial direction, assume <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> is weakly locally connected. To show it is locally connected, it is enough to show that the <a href="/wiki/Connected_component_(topology)" class="mw-redirect" title="Connected component (topology)">connected components</a> of open sets are open. </p><p>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></span> be open in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> and let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span> be a connected component of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a305ef479ab152035f334467a2c314baa23eb36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.429ex; height:2.176ex;" alt="{\displaystyle U.}"></span> Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> be an element of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/067be67e68f60c53ce83241748d0d6249675c58d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.413ex; height:2.176ex;" alt="{\displaystyle C.}"></span> Then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></span> is a neighborhood of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> so that there is a connected neighborhood <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> contained in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a305ef479ab152035f334467a2c314baa23eb36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.429ex; height:2.176ex;" alt="{\displaystyle U.}"></span> Since <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> is connected and contains <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/feff4d40084c7351bf57b11ba2427f6331f5bdbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.977ex; height:2.009ex;" alt="{\displaystyle x,}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> must be a subset of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span> (the connected component containing <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>). Therefore <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> is an interior point of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/067be67e68f60c53ce83241748d0d6249675c58d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.413ex; height:2.176ex;" alt="{\displaystyle C.}"></span> Since <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> was an arbitrary point of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64528f031cdbe1f52bdaf4ba7a8401108c0d2dc2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.413ex; height:2.509ex;" alt="{\displaystyle C,}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span> is open in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ba76c5a460c4a0bb1639a193bc1830f0a773e03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}"></span> Therefore, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> is locally connected. </p> </td></tr></tbody></table></div> <ol><li>Local connectedness is, by definition, a <a href="/wiki/Local_property" title="Local property">local property</a> of topological spaces, i.e., a topological property <i>P</i> such that a space <i>X</i> possesses property <i>P</i> if and only if each point <i>x</i> in <i>X</i> admits a neighborhood base of sets that have property <i>P</i>. Accordingly, all the "metaproperties" held by a local property hold for local connectedness. In particular:</li> <li>A space is locally connected if and only if it admits a <a href="/wiki/Base_(topology)" title="Base (topology)">base</a> of (open) connected subsets.</li> <li>The <a href="/wiki/Disjoint_union_(topology)" title="Disjoint union (topology)">disjoint union</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \coprod _{i}X_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∐</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \coprod _{i}X_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e45d683436bf181f0d885fbddaa48cc518d68029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:6.08ex; height:5.509ex;" alt="{\displaystyle \coprod _{i}X_{i}}"></span> of a family <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{X_{i}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{X_{i}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ba8a662382c1f121e79e3f1d518c575cb6c42c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.049ex; height:2.843ex;" alt="{\displaystyle \{X_{i}\}}"></span> of spaces is locally connected if and only if each <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af4a0955af42beb5f85aa05fb8c07abedc13990d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.724ex; height:2.509ex;" alt="{\displaystyle X_{i}}"></span> is locally connected. In particular, since a single point is certainly locally connected, it follows that any <a href="/wiki/Discrete_space" title="Discrete space">discrete space</a> is locally connected. On the other hand, a discrete space is <a href="/wiki/Totally_disconnected" class="mw-redirect" title="Totally disconnected">totally disconnected</a>, so is connected only if it has at most one point.</li> <li>Conversely, a <a href="/wiki/Totally_disconnected_space" title="Totally disconnected space">totally disconnected space</a> is locally connected if and only if it is discrete. This can be used to explain the aforementioned fact that the rational numbers are not locally connected.</li> <li>A nonempty product space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \prod _{i}X_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \prod _{i}X_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25951f7a02818dda8ba0c0a0224df04506bc6694" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:6.08ex; height:5.509ex;" alt="{\displaystyle \prod _{i}X_{i}}"></span> is locally connected if and only if each <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af4a0955af42beb5f85aa05fb8c07abedc13990d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.724ex; height:2.509ex;" alt="{\displaystyle X_{i}}"></span> is locally connected and all but finitely many of the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af4a0955af42beb5f85aa05fb8c07abedc13990d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.724ex; height:2.509ex;" alt="{\displaystyle X_{i}}"></span> are connected.<sup id="cite_ref-FOOTNOTEWillard2004201Theorem_27.13_15-0" class="reference"><a href="#cite_note-FOOTNOTEWillard2004201Theorem_27.13-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup></li> <li>Every <a href="/wiki/Hyperconnected_space" title="Hyperconnected space">hyperconnected space</a> is locally connected, and connected.</li></ol> <div class="mw-heading mw-heading2"><h2 id="Components_and_path_components">Components and path components</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Locally_connected_space&action=edit&section=6" title="Edit section: Components and path components"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The following result follows almost immediately from the definitions but will be quite useful: </p><p>Lemma: Let <i>X</i> be a space, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{Y_{i}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{Y_{i}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6e92004a07c56dd2b161c0647531c2893103b34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.475ex; height:2.843ex;" alt="{\displaystyle \{Y_{i}\}}"></span> a family of subsets of <i>X</i>. Suppose that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bigcap _{i}Y_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>⋂</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bigcap _{i}Y_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f01dbee29f4dc0ee1bee31dcbc59e39343b93935" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:5.119ex; height:5.509ex;" alt="{\displaystyle \bigcap _{i}Y_{i}}"></span> is nonempty. Then, if each <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d57be496fff95ee2a97ee43c7f7fe244b4dbf8ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.15ex; height:2.509ex;" alt="{\displaystyle Y_{i}}"></span> is connected (respectively, path connected) then the union <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bigcup _{i}Y_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bigcup _{i}Y_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d88ed01f068906dd96994b3ab54b1b18a6f2f85c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:5.119ex; height:5.509ex;" alt="{\displaystyle \bigcup _{i}Y_{i}}"></span> is connected (respectively, path connected).<sup id="cite_ref-FOOTNOTEWillard2004192Theorem_26.7a_16-0" class="reference"><a href="#cite_note-FOOTNOTEWillard2004192Theorem_26.7a-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p><p>Now consider two relations on a topological space <i>X</i>: for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y\in X,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mi>X</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y\in X,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10c8e2c619d7ee4f04b7422de1bf3ac3945c12ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.987ex; height:2.509ex;" alt="{\displaystyle x,y\in X,}"></span> write: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\equiv _{c}y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <msub> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\equiv _{c}y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/221f4ca200b803b73ff4776a4b5184c02033823d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.528ex; height:2.009ex;" alt="{\displaystyle x\equiv _{c}y}"></span> if there is a connected subset of <i>X</i> containing both <i>x</i> and <i>y</i>; and</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\equiv _{pc}y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <msub> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mi>c</mi> </mrow> </msub> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\equiv _{pc}y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e05a49fca6b3ebe5a31197caf07a3a5646e87b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.355ex; height:2.343ex;" alt="{\displaystyle x\equiv _{pc}y}"></span> if there is a path connected subset of <i>X</i> containing both <i>x</i> and <i>y</i>.</dd></dl> <p>Evidently both relations are reflexive and symmetric. Moreover, if <i>x</i> and <i>y</i> are contained in a connected (respectively, path connected) subset <i>A</i> and <i>y</i> and <i>z</i> are connected in a connected (respectively, path connected) subset <i>B</i>, then the Lemma implies that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\cup B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∪</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\cup B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdb575990bcfbcdf616aa6fd76e8b30bf7fd2169" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.09ex; height:2.176ex;" alt="{\displaystyle A\cup B}"></span> is a connected (respectively, path connected) subset containing <i>x</i>, <i>y</i> and <i>z</i>. Thus each relation is an <a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence relation</a>, and defines a partition of <i>X</i> into <a href="/wiki/Equivalence_classes" class="mw-redirect" title="Equivalence classes">equivalence classes</a>. We consider these two partitions in turn. </p><p>For <i>x</i> in <i>X</i>, the set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46c7c57eb7d5bafc9dbb07668033f00118f5e48e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.834ex; height:2.509ex;" alt="{\displaystyle C_{x}}"></span> of all points <i>y</i> such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\equiv _{c}x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <msub> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\equiv _{c}x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b54a6b515f9e231cc760d3d761451b9fead341fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.528ex; height:2.009ex;" alt="{\displaystyle y\equiv _{c}x}"></span> is called the <a href="/wiki/Connected_component_(topology)" class="mw-redirect" title="Connected component (topology)">connected component</a> of <i>x</i>.<sup id="cite_ref-FOOTNOTEWillard2004194Definition_26.11_17-0" class="reference"><a href="#cite_note-FOOTNOTEWillard2004194Definition_26.11-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> The Lemma implies that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46c7c57eb7d5bafc9dbb07668033f00118f5e48e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.834ex; height:2.509ex;" alt="{\displaystyle C_{x}}"></span> is the unique maximal connected subset of <i>X</i> containing <i>x</i>.<sup id="cite_ref-FOOTNOTEWillard2004195–196Problem_26B_18-0" class="reference"><a href="#cite_note-FOOTNOTEWillard2004195–196Problem_26B-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> Since the closure of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46c7c57eb7d5bafc9dbb07668033f00118f5e48e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.834ex; height:2.509ex;" alt="{\displaystyle C_{x}}"></span> is also a connected subset containing <i>x</i>,<sup id="cite_ref-FOOTNOTEKelley197554Theorem_20_19-0" class="reference"><a href="#cite_note-FOOTNOTEKelley197554Theorem_20-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTEWillard2004193Theorem_26.8_20-0" class="reference"><a href="#cite_note-FOOTNOTEWillard2004193Theorem_26.8-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> it follows that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46c7c57eb7d5bafc9dbb07668033f00118f5e48e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.834ex; height:2.509ex;" alt="{\displaystyle C_{x}}"></span> is closed.<sup id="cite_ref-FOOTNOTEWillard2004194Theorem_26.12_21-0" class="reference"><a href="#cite_note-FOOTNOTEWillard2004194Theorem_26.12-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p><p>If <i>X</i> has only finitely many connected components, then each component is the complement of a finite union of closed sets and therefore open. In general, the connected components need not be open, since, e.g., there exist totally disconnected spaces (i.e., <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{x}=\{x\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{x}=\{x\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/075fb52567164a91aa7cffda41a1541e0f5fed04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.587ex; height:2.843ex;" alt="{\displaystyle C_{x}=\{x\}}"></span> for all points <i>x</i>) that are not discrete, like Cantor space. However, the connected components of a locally connected space are also open, and thus are <a href="/wiki/Clopen_sets" class="mw-redirect" title="Clopen sets">clopen sets</a>.<sup id="cite_ref-FOOTNOTEWillard2004200Corollary_27.10_22-0" class="reference"><a href="#cite_note-FOOTNOTEWillard2004200Corollary_27.10-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> It follows that a locally connected space <i>X</i> is a topological disjoint union <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \coprod C_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∐</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \coprod C_{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5bf69996bc25747f31c5bea622742dc75997003" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:6.191ex; height:3.843ex;" alt="{\displaystyle \coprod C_{x}}"></span> of its distinct connected components. Conversely, if for every open subset <i>U</i> of <i>X</i>, the connected components of <i>U</i> are open, then <i>X</i> admits a base of connected sets and is therefore locally connected.<sup id="cite_ref-FOOTNOTEWillard2004200Theorem_27.9_23-0" class="reference"><a href="#cite_note-FOOTNOTEWillard2004200Theorem_27.9-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> </p><p>Similarly <i>x</i> in <i>X</i>, the set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle PC_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle PC_{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18b0faf5b127ea4b6c44670b8a5242eda4504412" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.58ex; height:2.509ex;" alt="{\displaystyle PC_{x}}"></span> of all points <i>y</i> such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\equiv _{pc}x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <msub> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mi>c</mi> </mrow> </msub> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\equiv _{pc}x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ee45307f3ebdf89dfb25a637dda0b1cd898e2d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.355ex; height:2.343ex;" alt="{\displaystyle y\equiv _{pc}x}"></span> is called the <i>path component</i> of <i>x</i>.<sup id="cite_ref-FOOTNOTEWillard2004202Problem_27D_24-0" class="reference"><a href="#cite_note-FOOTNOTEWillard2004202Problem_27D-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> As above, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle PC_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle PC_{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18b0faf5b127ea4b6c44670b8a5242eda4504412" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.58ex; height:2.509ex;" alt="{\displaystyle PC_{x}}"></span> is also the union of all path connected subsets of <i>X</i> that contain <i>x</i>, so by the Lemma is itself path connected. Because path connected sets are connected, we have <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle PC_{x}\subseteq C_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>⊆</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle PC_{x}\subseteq C_{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44876a9c811be131906e6eb8751bea1ebcf0349c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.512ex; height:2.509ex;" alt="{\displaystyle PC_{x}\subseteq C_{x}}"></span> for all <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>X</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in X.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0deab6a01578b5b543b772df12dc0d2c593cc924" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.797ex; height:2.176ex;" alt="{\displaystyle x\in X.}"></span> </p><p>However the closure of a path connected set need not be path connected: for instance, the topologist's sine curve is the closure of the open subset <i>U</i> consisting of all points <i>(x,sin(x))</i> with <i>x > 0</i>, and <i>U</i>, being homeomorphic to an interval on the real line, is certainly path connected. Moreover, the path components of the topologist's sine curve <i>C</i> are <i>U</i>, which is open but not closed, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C\setminus U,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo class="MJX-variant">∖</mo> <mi>U</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C\setminus U,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/663eb1d7c1a5d1ab27798bb585c0babe618a9f4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.39ex; height:2.843ex;" alt="{\displaystyle C\setminus U,}"></span> which is closed but not open. </p><p>A space is locally path connected if and only if for all open subsets <i>U</i>, the path components of <i>U</i> are open.<sup id="cite_ref-FOOTNOTEWillard2004202Problem_27D_24-1" class="reference"><a href="#cite_note-FOOTNOTEWillard2004202Problem_27D-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> Therefore the path components of a locally path connected space give a partition of <i>X</i> into pairwise disjoint open sets. It follows that an open connected subspace of a locally path connected space is necessarily path connected.<sup id="cite_ref-FOOTNOTEWillard2004199Theorem_27.5_25-0" class="reference"><a href="#cite_note-FOOTNOTEWillard2004199Theorem_27.5-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> Moreover, if a space is locally path connected, then it is also locally connected, so for all <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>X</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in X,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3ebdb0a09f0721ccdd0b779e0a21caf386be82a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.797ex; height:2.509ex;" alt="{\displaystyle x\in X,}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46c7c57eb7d5bafc9dbb07668033f00118f5e48e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.834ex; height:2.509ex;" alt="{\displaystyle C_{x}}"></span> is connected and open, hence path connected, that is, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{x}=PC_{x}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mi>P</mi> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{x}=PC_{x}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6effb0769f69a69fc8b38b7ee0235da160d098f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.159ex; height:2.509ex;" alt="{\displaystyle C_{x}=PC_{x}.}"></span> That is, for a locally path connected space the components and path components coincide. </p> <div class="mw-heading mw-heading3"><h3 id="Examples">Examples</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Locally_connected_space&action=edit&section=7" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ol><li>The set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I\times I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo>×</mo> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I\times I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e290ebbc366ef6f8dd83eb6c3d5063f983f1783d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.184ex; height:2.176ex;" alt="{\displaystyle I\times I}"></span> (where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I=[0,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo>=</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I=[0,1]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87ec65159c44769434523e46928bc1b82681f842" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.923ex; height:2.843ex;" alt="{\displaystyle I=[0,1]}"></span>) in the <a href="/wiki/Lexicographical_order" class="mw-redirect" title="Lexicographical order">dictionary</a> <a href="/wiki/Order_topology" title="Order topology">order topology</a> has exactly one component (because it is connected) but has uncountably many path components. Indeed, any set of the form <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{a\}\times I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>a</mi> <mo fence="false" stretchy="false">}</mo> <mo>×</mo> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{a\}\times I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8da0c7c39892e9ef120c1c4d512633b3959b7df7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.567ex; height:2.843ex;" alt="{\displaystyle \{a\}\times I}"></span> is a path component for each <i>a</i> belonging to <i>I</i>.</li> <li>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:\mathbb {R} \to \mathbb {R} _{\ell }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">→</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:\mathbb {R} \to \mathbb {R} _{\ell }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/feb4356b124df28a349cae0a73a2dc1f409d77e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.104ex; height:2.509ex;" alt="{\displaystyle f:\mathbb {R} \to \mathbb {R} _{\ell }}"></span> be a continuous map from <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span> to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} _{\ell }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} _{\ell }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8da5ca620fe8ab904e3f95f580e439e01cbbce6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.596ex; height:2.509ex;" alt="{\displaystyle \mathbb {R} _{\ell }}"></span> (which is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span> in the <a href="/wiki/Lower_limit_topology" title="Lower limit topology">lower limit topology</a>). Since <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span> is connected, and the image of a connected space under a continuous map must be connected, the image of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span> under <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> must be connected. Therefore, the image of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span> under <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> must be a subset of a component of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} _{\ell }/}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} _{\ell }/}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/126f8ca47ceefd48c7e718c500f5318f9afb00f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.758ex; height:2.843ex;" alt="{\displaystyle \mathbb {R} _{\ell }/}"></span> Since this image is nonempty, the only continuous maps from '<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span> to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} _{\ell },}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} _{\ell },}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ef4b62e7cede23d374f33c364aecd6dc7b80dc3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.243ex; height:2.509ex;" alt="{\displaystyle \mathbb {R} _{\ell },}"></span> are the constant maps. In fact, any continuous map from a connected space to a totally disconnected space must be constant.</li></ol> <div class="mw-heading mw-heading2"><h2 id="Quasicomponents">Quasicomponents</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Locally_connected_space&action=edit&section=8" title="Edit section: Quasicomponents"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <i>X</i> be a topological space. We define a third relation on <i>X</i>: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\equiv _{qc}y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <msub> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> <mi>c</mi> </mrow> </msub> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\equiv _{qc}y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83e702082f20e2d6a3dcd82f38109c3798767378" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.284ex; height:2.343ex;" alt="{\displaystyle x\equiv _{qc}y}"></span> if there is no separation of <i>X</i> into open sets <i>A</i> and <i>B</i> such that <i>x</i> is an element of <i>A</i> and <i>y</i> is an element of <i>B</i>. This is an equivalence relation on <i>X</i> and the equivalence class <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle QC_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle QC_{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a394ae99b4479eafdb31fad0a22dea8bb593afd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.673ex; height:2.509ex;" alt="{\displaystyle QC_{x}}"></span> containing <i>x</i> is called the <b>quasicomponent</b> of <i>x</i>.<sup id="cite_ref-FOOTNOTEWillard2004195–196Problem_26B_18-1" class="reference"><a href="#cite_note-FOOTNOTEWillard2004195–196Problem_26B-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle QC_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle QC_{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a394ae99b4479eafdb31fad0a22dea8bb593afd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.673ex; height:2.509ex;" alt="{\displaystyle QC_{x}}"></span> can also be characterized as the intersection of all <a href="/wiki/Clopen" class="mw-redirect" title="Clopen">clopen</a> subsets of <i>X</i> that contain <i>x</i>.<sup id="cite_ref-FOOTNOTEWillard2004195–196Problem_26B_18-2" class="reference"><a href="#cite_note-FOOTNOTEWillard2004195–196Problem_26B-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> Accordingly <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle QC_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle QC_{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a394ae99b4479eafdb31fad0a22dea8bb593afd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.673ex; height:2.509ex;" alt="{\displaystyle QC_{x}}"></span> is closed; in general it need not be open. </p><p>Evidently <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{x}\subseteq QC_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>⊆</mo> <mi>Q</mi> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{x}\subseteq QC_{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d4e49059450c87fa6b579f0b999a9717daa4d38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.605ex; height:2.509ex;" alt="{\displaystyle C_{x}\subseteq QC_{x}}"></span> for all <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>X</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in X.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0deab6a01578b5b543b772df12dc0d2c593cc924" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.797ex; height:2.176ex;" alt="{\displaystyle x\in X.}"></span><sup id="cite_ref-FOOTNOTEWillard2004195–196Problem_26B_18-3" class="reference"><a href="#cite_note-FOOTNOTEWillard2004195–196Problem_26B-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> Overall we have the following containments among path components, components and quasicomponents at <i>x</i>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle PC_{x}\subseteq C_{x}\subseteq QC_{x}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>⊆</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>⊆</mo> <mi>Q</mi> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle PC_{x}\subseteq C_{x}\subseteq QC_{x}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee358b9515a33c56c5c0ed2e041a66b1ed63d0b7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.93ex; height:2.509ex;" alt="{\displaystyle PC_{x}\subseteq C_{x}\subseteq QC_{x}.}"></span> </p><p>If <i>X</i> is locally connected, then, as above, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46c7c57eb7d5bafc9dbb07668033f00118f5e48e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.834ex; height:2.509ex;" alt="{\displaystyle C_{x}}"></span> is a clopen set containing <i>x</i>, so <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle QC_{x}\subseteq C_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>⊆</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle QC_{x}\subseteq C_{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74d5e990a63b2fea5155bc3dfe97e5211bc1a671" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.605ex; height:2.509ex;" alt="{\displaystyle QC_{x}\subseteq C_{x}}"></span> and thus <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle QC_{x}=C_{x}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle QC_{x}=C_{x}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2dacc9c988b4c9602fe39144ac81b80e8f072380" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.252ex; height:2.509ex;" alt="{\displaystyle QC_{x}=C_{x}.}"></span> Since local path connectedness implies local connectedness, it follows that at all points <i>x</i> of a locally path connected space we have <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle PC_{x}=C_{x}=QC_{x}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mi>Q</mi> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle PC_{x}=C_{x}=QC_{x}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/697ed04db1d4e1ddab566371812d5c41b113ef9c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.93ex; height:2.509ex;" alt="{\displaystyle PC_{x}=C_{x}=QC_{x}.}"></span> </p><p>Another class of spaces for which the quasicomponents agree with the components is the class of compact Hausdorff spaces.<sup id="cite_ref-FOOTNOTEEngelking1989357Theorem_6.1.23_26-0" class="reference"><a href="#cite_note-FOOTNOTEEngelking1989357Theorem_6.1.23-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Examples_2">Examples</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Locally_connected_space&action=edit&section=9" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ol><li>An example of a space whose quasicomponents are not equal to its components is a sequence with a double limit point. This space is totally disconnected, but both limit points lie in the same quasicomponent, because any clopen set containing one of them must contain a tail of the sequence, and thus the other point too.</li> <li>The space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\{0\}\cup \{{\frac {1}{n}}:n\in \mathbb {Z} ^{+}\})\times [-1,1]\setminus \{(0,0)\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo>∪</mo> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <mo>:</mo> <mi>n</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">)</mo> <mo>×</mo> <mo stretchy="false">[</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo class="MJX-variant">∖</mo> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\{0\}\cup \{{\frac {1}{n}}:n\in \mathbb {Z} ^{+}\})\times [-1,1]\setminus \{(0,0)\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4cc928a81f0fed04952ee17e3956aeae7046035" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:40.658ex; height:5.176ex;" alt="{\displaystyle (\{0\}\cup \{{\frac {1}{n}}:n\in \mathbb {Z} ^{+}\})\times [-1,1]\setminus \{(0,0)\}}"></span> is locally compact and Hausdorff but the sets <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{0\}\times [-1,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo>×</mo> <mo stretchy="false">[</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{0\}\times [-1,0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00521f70fa6e924663bf0590d2e0734b4ddee02a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.046ex; height:2.843ex;" alt="{\displaystyle \{0\}\times [-1,0)}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{0\}\times (0,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo>×</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{0\}\times (0,1]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01c9697d05fbb7c7c383fea878ae14a4c7849e77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.238ex; height:2.843ex;" alt="{\displaystyle \{0\}\times (0,1]}"></span> are two different components which lie in the same quasicomponent.</li> <li>The <a href="/wiki/Arens%E2%80%93Fort_space" title="Arens–Fort space">Arens–Fort space</a> is not locally connected, but nevertheless the components and the quasicomponents coincide: indeed <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle QC_{x}=C_{x}=\{x\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle QC_{x}=C_{x}=\{x\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8cd77869fd525dbb8933c7fd8a39d0f121eb38d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.358ex; height:2.843ex;" alt="{\displaystyle QC_{x}=C_{x}=\{x\}}"></span> for all points <i>x</i>.<sup id="cite_ref-FOOTNOTESteenSeebach199554–55_27-0" class="reference"><a href="#cite_note-FOOTNOTESteenSeebach199554–55-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup></li></ol> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Locally_connected_space&action=edit&section=10" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Locally_simply_connected_space" title="Locally simply connected space">Locally simply connected space</a></li> <li><a href="/wiki/Semi-locally_simply_connected" title="Semi-locally simply connected">Semi-locally simply connected</a></li> <li><a href="/wiki/MLC_conjecture" class="mw-redirect" title="MLC conjecture">It is conjectured that the Mandelbrot set is locally connected</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Locally_connected_space&action=edit&section=11" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-FOOTNOTEMunkres2000161-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEMunkres2000161_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEMunkres2000161_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-FOOTNOTEMunkres2000161_1-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-FOOTNOTEMunkres2000161_1-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFMunkres2000">Munkres 2000</a>, p. 161.</span> </li> <li id="cite_note-FOOTNOTEWillard2004199Definition_27.7-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEWillard2004199Definition_27.7_2-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFWillard2004">Willard 2004</a>, p. 199, Definition 27.7.<span class="error harv-error" style="display: none; font-size:100%"> sfn error: no target: CITEREFWillard2004 (<a href="/wiki/Category:Harv_and_Sfn_template_errors" title="Category:Harv and Sfn template errors">help</a>)</span></span> </li> <li id="cite_note-FOOTNOTEWillard2004199Definition_27.4-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEWillard2004199Definition_27.4_3-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFWillard2004">Willard 2004</a>, p. 199, Definition 27.4.<span class="error harv-error" style="display: none; font-size:100%"> sfn error: no target: CITEREFWillard2004 (<a href="/wiki/Category:Harv_and_Sfn_template_errors" title="Category:Harv and Sfn template errors">help</a>)</span></span> </li> <li id="cite_note-FOOTNOTEWillard2004201Definition_27.14-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEWillard2004201Definition_27.14_4-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFWillard2004">Willard 2004</a>, p. 201, Definition 27.14.<span class="error harv-error" style="display: none; font-size:100%"> sfn error: no target: CITEREFWillard2004 (<a href="/wiki/Category:Harv_and_Sfn_template_errors" title="Category:Harv and Sfn template errors">help</a>)</span></span> </li> <li id="cite_note-BBS-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-BBS_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-BBS_5-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFBjörnBjörnShanmugalingam2016" class="citation journal cs1">Björn, Anders; Björn, Jana; Shanmugalingam, Nageswari (2016). "The Mazurkiewicz distance and sets that are finitely connected at the boundary". <i>Journal of Geometric Analysis</i>. <b>26</b> (2): 873–897. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1311.5122">1311.5122</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs12220-015-9575-9">10.1007/s12220-015-9575-9</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:255549682">255549682</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Geometric+Analysis&rft.atitle=The+Mazurkiewicz+distance+and+sets+that+are+finitely+connected+at+the+boundary&rft.volume=26&rft.issue=2&rft.pages=873-897&rft.date=2016&rft_id=info%3Aarxiv%2F1311.5122&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A255549682%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2Fs12220-015-9575-9&rft.aulast=Bj%C3%B6rn&rft.aufirst=Anders&rft.au=Bj%C3%B6rn%2C+Jana&rft.au=Shanmugalingam%2C+Nageswari&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALocally+connected+space" class="Z3988"></span>, section 2</span> </li> <li id="cite_note-FOOTNOTEMunkres2000162Exercise_6-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEMunkres2000162Exercise_6_6-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFMunkres2000">Munkres 2000</a>, p. 162, Exercise 6.</span> </li> <li id="cite_note-FOOTNOTESteenSeebach1995139Example_119.4-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTESteenSeebach1995139Example_119.4_7-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFSteenSeebach1995">Steen & Seebach 1995</a>, p. 139, Example 119.4.</span> </li> <li id="cite_note-FOOTNOTEMunkres2000162Exercise_7-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEMunkres2000162Exercise_7_8-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFMunkres2000">Munkres 2000</a>, p. 162, Exercise 7.</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://math.stackexchange.com/q/2439096">"Show that X is not locally connected at p"</a>. <i>Math StackExchange</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Math+StackExchange&rft.atitle=Show+that+X+is+not+locally+connected+at+p&rft_id=https%3A%2F%2Fmath.stackexchange.com%2Fq%2F2439096&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALocally+connected+space" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTEWillard2004201Theorem_27.16-10"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEWillard2004201Theorem_27.16_10-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEWillard2004201Theorem_27.16_10-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFWillard2004">Willard 2004</a>, p. 201, Theorem 27.16.<span class="error harv-error" style="display: none; font-size:100%"> sfn error: no target: CITEREFWillard2004 (<a href="/wiki/Category:Harv_and_Sfn_template_errors" title="Category:Harv and Sfn template errors">help</a>)</span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://math.stackexchange.com/q/2999685">"Definition of locally pathwise connected"</a>. <i>Math StackExchange</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Math+StackExchange&rft.atitle=Definition+of+locally+pathwise+connected&rft_id=https%3A%2F%2Fmath.stackexchange.com%2Fq%2F2999685&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALocally+connected+space" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTESteenSeebach1995137–138-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTESteenSeebach1995137–138_12-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFSteenSeebach1995">Steen & Seebach 1995</a>, pp. 137–138.</span> </li> <li id="cite_note-FOOTNOTESteenSeebach199549–50-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTESteenSeebach199549–50_13-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFSteenSeebach1995">Steen & Seebach 1995</a>, pp. 49–50.</span> </li> <li id="cite_note-FOOTNOTESteenSeebach199573Example_48-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTESteenSeebach199573Example_48_14-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFSteenSeebach1995">Steen & Seebach 1995</a>, p. 73, Example 48.</span> </li> <li id="cite_note-FOOTNOTEWillard2004201Theorem_27.13-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEWillard2004201Theorem_27.13_15-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFWillard2004">Willard 2004</a>, p. 201, Theorem 27.13.<span class="error harv-error" style="display: none; font-size:100%"> sfn error: no target: CITEREFWillard2004 (<a href="/wiki/Category:Harv_and_Sfn_template_errors" title="Category:Harv and Sfn template errors">help</a>)</span></span> </li> <li id="cite_note-FOOTNOTEWillard2004192Theorem_26.7a-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEWillard2004192Theorem_26.7a_16-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFWillard2004">Willard 2004</a>, p. 192, Theorem 26.7a.<span class="error harv-error" style="display: none; font-size:100%"> sfn error: no target: CITEREFWillard2004 (<a href="/wiki/Category:Harv_and_Sfn_template_errors" title="Category:Harv and Sfn template errors">help</a>)</span></span> </li> <li id="cite_note-FOOTNOTEWillard2004194Definition_26.11-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEWillard2004194Definition_26.11_17-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFWillard2004">Willard 2004</a>, p. 194, Definition 26.11.<span class="error harv-error" style="display: none; font-size:100%"> sfn error: no target: CITEREFWillard2004 (<a href="/wiki/Category:Harv_and_Sfn_template_errors" title="Category:Harv and Sfn template errors">help</a>)</span></span> </li> <li id="cite_note-FOOTNOTEWillard2004195–196Problem_26B-18"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEWillard2004195–196Problem_26B_18-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEWillard2004195–196Problem_26B_18-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-FOOTNOTEWillard2004195–196Problem_26B_18-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-FOOTNOTEWillard2004195–196Problem_26B_18-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFWillard2004">Willard 2004</a>, pp. 195–196, Problem 26B.<span class="error harv-error" style="display: none; font-size:100%"> sfn error: no target: CITEREFWillard2004 (<a href="/wiki/Category:Harv_and_Sfn_template_errors" title="Category:Harv and Sfn template errors">help</a>)</span></span> </li> <li id="cite_note-FOOTNOTEKelley197554Theorem_20-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEKelley197554Theorem_20_19-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFKelley1975">Kelley 1975</a>, p. 54, Theorem 20.<span class="error harv-error" style="display: none; font-size:100%"> sfn error: no target: CITEREFKelley1975 (<a href="/wiki/Category:Harv_and_Sfn_template_errors" title="Category:Harv and Sfn template errors">help</a>)</span></span> </li> <li id="cite_note-FOOTNOTEWillard2004193Theorem_26.8-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEWillard2004193Theorem_26.8_20-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFWillard2004">Willard 2004</a>, p. 193, Theorem 26.8.<span class="error harv-error" style="display: none; font-size:100%"> sfn error: no target: CITEREFWillard2004 (<a href="/wiki/Category:Harv_and_Sfn_template_errors" title="Category:Harv and Sfn template errors">help</a>)</span></span> </li> <li id="cite_note-FOOTNOTEWillard2004194Theorem_26.12-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEWillard2004194Theorem_26.12_21-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFWillard2004">Willard 2004</a>, p. 194, Theorem 26.12.<span class="error harv-error" style="display: none; font-size:100%"> sfn error: no target: CITEREFWillard2004 (<a href="/wiki/Category:Harv_and_Sfn_template_errors" title="Category:Harv and Sfn template errors">help</a>)</span></span> </li> <li id="cite_note-FOOTNOTEWillard2004200Corollary_27.10-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEWillard2004200Corollary_27.10_22-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFWillard2004">Willard 2004</a>, p. 200, Corollary 27.10.<span class="error harv-error" style="display: none; font-size:100%"> sfn error: no target: CITEREFWillard2004 (<a href="/wiki/Category:Harv_and_Sfn_template_errors" title="Category:Harv and Sfn template errors">help</a>)</span></span> </li> <li id="cite_note-FOOTNOTEWillard2004200Theorem_27.9-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEWillard2004200Theorem_27.9_23-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFWillard2004">Willard 2004</a>, p. 200, Theorem 27.9.<span class="error harv-error" style="display: none; font-size:100%"> sfn error: no target: CITEREFWillard2004 (<a href="/wiki/Category:Harv_and_Sfn_template_errors" title="Category:Harv and Sfn template errors">help</a>)</span></span> </li> <li id="cite_note-FOOTNOTEWillard2004202Problem_27D-24"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEWillard2004202Problem_27D_24-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEWillard2004202Problem_27D_24-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFWillard2004">Willard 2004</a>, p. 202, Problem 27D.<span class="error harv-error" style="display: none; font-size:100%"> sfn error: no target: CITEREFWillard2004 (<a href="/wiki/Category:Harv_and_Sfn_template_errors" title="Category:Harv and Sfn template errors">help</a>)</span></span> </li> <li id="cite_note-FOOTNOTEWillard2004199Theorem_27.5-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEWillard2004199Theorem_27.5_25-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFWillard2004">Willard 2004</a>, p. 199, Theorem 27.5.<span class="error harv-error" style="display: none; font-size:100%"> sfn error: no target: CITEREFWillard2004 (<a href="/wiki/Category:Harv_and_Sfn_template_errors" title="Category:Harv and Sfn template errors">help</a>)</span></span> </li> <li id="cite_note-FOOTNOTEEngelking1989357Theorem_6.1.23-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEEngelking1989357Theorem_6.1.23_26-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFEngelking1989">Engelking 1989</a>, p. 357, Theorem 6.1.23.</span> </li> <li id="cite_note-FOOTNOTESteenSeebach199554–55-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTESteenSeebach199554–55_27-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFSteenSeebach1995">Steen & Seebach 1995</a>, pp. 54–55.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Locally_connected_space&action=edit&section=12" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEngelking1989" class="citation book cs1"><a href="/wiki/Ryszard_Engelking" title="Ryszard Engelking">Engelking, Ryszard</a> (1989). <i>General Topology</i>. Heldermann Verlag, Berlin. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-88538-006-4" title="Special:BookSources/3-88538-006-4"><bdi>3-88538-006-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=General+Topology&rft.pub=Heldermann+Verlag%2C+Berlin&rft.date=1989&rft.isbn=3-88538-006-4&rft.aulast=Engelking&rft.aufirst=Ryszard&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALocally+connected+space" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKelley1975" class="citation book cs1"><a href="/wiki/John_L._Kelley" title="John L. Kelley">Kelley, John L.</a> (1975) [1955]. <i>General Topology</i>. <a href="/wiki/Graduate_Texts_in_Mathematics" title="Graduate Texts in Mathematics">Graduate Texts in Mathematics</a>. Vol. 27 (2nd ed.). New York: Springer-Verlag. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-90125-1" title="Special:BookSources/978-0-387-90125-1"><bdi>978-0-387-90125-1</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/1365153">1365153</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=General+Topology&rft.place=New+York&rft.series=Graduate+Texts+in+Mathematics&rft.edition=2nd&rft.pub=Springer-Verlag&rft.date=1975&rft_id=info%3Aoclcnum%2F1365153&rft.isbn=978-0-387-90125-1&rft.aulast=Kelley&rft.aufirst=John+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALocally+connected+space" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMunkres2000" class="citation book cs1"><a href="/wiki/James_Munkres" title="James Munkres">Munkres, James R.</a> (2000). <i>Topology</i> (2nd ed.). <a href="/wiki/Upper_Saddle_River,_NJ" class="mw-redirect" title="Upper Saddle River, NJ">Upper Saddle River, NJ</a>: <a href="/wiki/Prentice_Hall,_Inc" class="mw-redirect" title="Prentice Hall, Inc">Prentice Hall, Inc</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-13-181629-9" title="Special:BookSources/978-0-13-181629-9"><bdi>978-0-13-181629-9</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/42683260">42683260</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Topology&rft.place=Upper+Saddle+River%2C+NJ&rft.edition=2nd&rft.pub=Prentice+Hall%2C+Inc&rft.date=2000&rft_id=info%3Aoclcnum%2F42683260&rft.isbn=978-0-13-181629-9&rft.aulast=Munkres&rft.aufirst=James+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALocally+connected+space" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSteenSeebach1995" class="citation cs2"><a href="/wiki/Lynn_Arthur_Steen" class="mw-redirect" title="Lynn Arthur Steen">Steen, Lynn Arthur</a>; <a href="/wiki/J._Arthur_Seebach,_Jr." class="mw-redirect" title="J. Arthur Seebach, Jr.">Seebach, J. Arthur Jr.</a> (1995) [1978], <i><a href="/wiki/Counterexamples_in_Topology" title="Counterexamples in Topology">Counterexamples in Topology</a></i> (<a href="/wiki/Dover_Publications" title="Dover Publications">Dover</a> reprint of 1978 ed.), Mineola, NY: Dover Publications, Inc., <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-68735-3" title="Special:BookSources/978-0-486-68735-3"><bdi>978-0-486-68735-3</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1382863">1382863</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Counterexamples+in+Topology&rft.place=Mineola%2C+NY&rft.edition=Dover+reprint+of+1978&rft.pub=Dover+Publications%2C+Inc.&rft.date=1995&rft.isbn=978-0-486-68735-3&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1382863%23id-name%3DMR&rft.aulast=Steen&rft.aufirst=Lynn+Arthur&rft.au=Seebach%2C+J.+Arthur+Jr.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALocally+connected+space" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWillard2004" class="citation book cs1">Willard, Stephen (2004) [1970]. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=-o8xJQ7Ag2cC"><i>General Topology</i></a>. <a href="/wiki/Mineola,_N.Y." class="mw-redirect" title="Mineola, N.Y.">Mineola, N.Y.</a>: <a href="/wiki/Dover_Publications" title="Dover Publications">Dover Publications</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-43479-7" title="Special:BookSources/978-0-486-43479-7"><bdi>978-0-486-43479-7</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/115240">115240</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=General+Topology&rft.place=Mineola%2C+N.Y.&rft.pub=Dover+Publications&rft.date=2004&rft_id=info%3Aoclcnum%2F115240&rft.isbn=978-0-486-43479-7&rft.aulast=Willard&rft.aufirst=Stephen&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D-o8xJQ7Ag2cC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALocally+connected+space" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Locally_connected_space&action=edit&section=13" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCoppin1972" class="citation cs2">Coppin, C. A. (1972), "Continuous Functions from a Connected Locally Connected Space into a Connected Space with a Dispersion Point", <i>Proceedings of the American Mathematical Society</i>, <b>32</b> (2), American Mathematical Society: 625–626, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0002-9939-1972-0296913-7">10.1090/S0002-9939-1972-0296913-7</a></span>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2037874">2037874</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+American+Mathematical+Society&rft.atitle=Continuous+Functions+from+a+Connected+Locally+Connected+Space+into+a+Connected+Space+with+a+Dispersion+Point&rft.volume=32&rft.issue=2&rft.pages=625-626&rft.date=1972&rft_id=info%3Adoi%2F10.1090%2FS0002-9939-1972-0296913-7&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2037874%23id-name%3DJSTOR&rft.aulast=Coppin&rft.aufirst=C.+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALocally+connected+space" class="Z3988"></span>. For Hausdorff spaces, it is shown that any continuous function from a connected locally connected space into a connected space with a dispersion point is constant</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDavis1968" class="citation cs2">Davis, H. S. (1968), "A Note on Connectedness Im Kleinen", <i>Proceedings of the American Mathematical Society</i>, <b>19</b> (5), American Mathematical Society: 1237–1241, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1090%2Fs0002-9939-1968-0254814-3">10.1090/s0002-9939-1968-0254814-3</a></span>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2036067">2036067</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+American+Mathematical+Society&rft.atitle=A+Note+on+Connectedness+Im+Kleinen&rft.volume=19&rft.issue=5&rft.pages=1237-1241&rft.date=1968&rft_id=info%3Adoi%2F10.1090%2Fs0002-9939-1968-0254814-3&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2036067%23id-name%3DJSTOR&rft.aulast=Davis&rft.aufirst=H.+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALocally+connected+space" class="Z3988"></span>.</li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Locally_connected_space&oldid=1260720003#locally_path_connected">https://en.wikipedia.org/w/index.php?title=Locally_connected_space&oldid=1260720003#locally_path_connected</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Properties_of_topological_spaces" title="Category:Properties of topological spaces">Properties of topological spaces</a></li><li><a href="/wiki/Category:General_topology" title="Category:General topology">General topology</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Harv_and_Sfn_no-target_errors" title="Category:Harv and Sfn no-target errors">Harv and Sfn no-target errors</a></li><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_matches_Wikidata" title="Category:Short description matches Wikidata">Short description matches Wikidata</a></li><li><a href="/wiki/Category:All_articles_lacking_reliable_references" title="Category:All articles lacking reliable references">All articles lacking reliable references</a></li><li><a href="/wiki/Category:Articles_lacking_reliable_references_from_December_2024" title="Category:Articles lacking reliable references from December 2024">Articles lacking reliable references from December 2024</a></li><li><a href="/wiki/Category:Articles_containing_proofs" title="Category:Articles containing proofs">Articles containing proofs</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 2 December 2024, at 06:43<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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