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<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg-flat.dtd" > <html xmlns="http://www.w3.org/1999/xhtml"> <head> <title> semi-locally simply-connected topological space in nLab </title> <meta http-equiv="Content-Type" content="text/html; charset=UTF-8" /> <meta name="robots" content="noindex,nofollow" /> <style type="text/css"> h1#pageName, div.info, .newWikiWord a, a.existingWikiWord, .newWikiWord a:hover, [actiontype="toggle"]:hover, #TextileHelp h3 { color: #226622; } </style> <link href="/stylesheets/instiki.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <style type="text/css"><![CDATA[/*></style> <script src="/javascripts/prototype.js?1660229990" type="text/javascript"></script> <script src="/javascripts/effects.js?1660229990" type="text/javascript"></script> <script src="/javascripts/dragdrop.js?1660229990" type="text/javascript"></script> <script src="/javascripts/controls.js?1660229990" type="text/javascript"></script> <script src="/javascripts/application.js?1660229990" type="text/javascript"></script> <script type="text/javascript"> <![CDATA[//><!]]> </script> </head> <body> <div id="Container"> <textarea id='content' readonly=' readonly' rows='24' cols='60' > +-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ### Context #### Topology +--{: .hide} [[!include topology - contents]] =-- =-- =-- #Contents# * automatic table of contents goes here {:toc} ## Idea A [[topological space]] $X$ is _(semi-)locally simply connected_ if every [[neighborhood]] of a point has a subneighbourhood in which [[loops]] based at the point in the subneighborhood can be contracted in $X$. It is similar to but weaker than the condition that every neighborhood of a point has a subneighborhood that is [[simply connected]]. This latter condition is called _local simple-connectedness_. ## Definition A [[topological space]] $X$ is **semi-locally simply-connected** if it has a basis of [[neighbourhood]]s $U$ such that the inclusion $\Pi_1(U) \to \Pi_1(X)$ of [[fundamental groupoid]]s factors through the canonical functor $\Pi_1(U) \to codisc(U)$ to the [[codiscrete groupoid]] whose objects are the elements of $U$. The condition on $U$ is equivalent to the condition that the homomorphism $\pi_1(U, x) \to \pi_1(X, x)$ of [[fundamental group]]s induced by inclusion $U \subseteq X$ is trivial. * A semi-locally simply connected space need not be locally simply connected. For a simple counterexample, take the cone on the [[Hawaiian earring space]]. ## Examples +-- {: .num_example #LocallySimplyConnectedCircle} ###### Example **([[circle]] is locally simply connected)** The [[Euclidean space|Euclidean]] [[circle]] $$ S^1 \;=\; \big\{ x \in \mathbb{R}^2 \;\big\vert\; {\Vert x\Vert} = 1 \big\} \;\subset\; \mathbb{R}^2 $$ is locally simply connected =-- +-- {: .proof} ###### Proof By definition of the [[subspace topology]] and the defining [[topological base]] of the [[Euclidean plane]], a [[base for a topology|base for the topology]] of $S^1$ is given by the [[images]] of [[open intervals]] under the [[local homeomorphism]] $$ \big(cos(-), sin(-)\big) \;\colon\; \mathbb{R}^1 \longrightarrow S^1 \,. $$ But these open intervals are simply connected ([this example](fundamental+group#EuclideanSpaceFundamentalGroup)). =-- * A semi-locally simply connected space need not be locally simply connected. For a simple counterexample, take the cone on the [[Hawaiian earring space]]. ## Application Semi-local simple connectedness is the crucial condition needed to have a good theory of [[covering space]]s, to the effect that the topos of permutation representations of the fundamental groupoid of $X$ is equivalent to the category of covering spaces of $X$. This is the _[[fundamental theorem of covering spaces]]_, see there for more. ## In topos theory For a [[topos]]-theoretic notion of locally $n$-connected see [[locally n-connected (infinity,1)-topos]]. ## Related concepts * [[locally path-connected topological space]] [[!redirects semi-locally simply-connected topological spaces]] [[!redirects semi-locally simply connected space]] [[!redirects semi-locally simply connected spaces]] [[!redirects semi-locally simply-connected space]] [[!redirects semi-locally simply-connected spaces]] [[!redirects semi-locally simply connected topological space]] [[!redirects semi-locally simply connected topological spaces]] [[!redirects semi-locally simply-connected topological space]] [[!redirects semi-locally simply-connected topological spaces]] [[!redirects locally simply connected space]] [[!redirects locally simply connected spaces]] [[!redirects locally simply-connected space]] [[!redirects locally simply-connected spaces]] [[!redirects locally simply connected topological space]] [[!redirects locally simply connected topological spaces]] [[!redirects locally simply-connected topological space]] [[!redirects locally simply-connected topological spaces]] [[!redirects semi-locally simply topological connected space]] </textarea> </div>  </body> </html>