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width: 0.3em;"></span> <a href="/nlab/show/diff/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/11088/#Item_1" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <p class="show_diff"> Showing changes from revision #1 to #2: <ins class="diffins">Added</ins> | <del class="diffdel">Removed</del> | <del class="diffmod">Chan</del><ins class="diffmod">ged</ins> </p> <h2 id='statement'>Statement</h2> <p>Recall that a <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a> <math class='maruku-mathml' display='inline' id='mathml_44f470209ecd7206b094a1bff11a171ba68ae7a1_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is <a class='existingWikiWord' href='/nlab/show/diff/paracompact+topological+space'>paracompact</a><span> if every<del class='diffdel'> open</del><del class='diffdel'> cover</del><del class='diffdel'> has</del><del class='diffdel'> a</del></span><ins class='diffins'><a class='existingWikiWord' href='/nlab/show/diff/open+cover'>open cover</a></ins><ins class='diffins'> has a </ins><a class='existingWikiWord' href='/nlab/show/diff/refinement'>refinement</a> by a <a class='existingWikiWord' href='/nlab/show/diff/locally+finite+cover'>locally finite open cover</a>. Further a space is called <a class='existingWikiWord' href='/nlab/show/diff/Lindel%C3%B6f+topological+space'>Lindelöf</a> if every <a class='existingWikiWord' href='/nlab/show/diff/open+cover'>open cover</a> has a <a class='existingWikiWord' href='/nlab/show/diff/countable+cover'>countable</a> sub-cover.</p> <p>\begin{theorem} Assuming the <a class='existingWikiWord' href='/nlab/show/diff/axiom+of+choice'>axiom of choice</a>:</p> <p>Every paracompact space <math class='maruku-mathml' display='inline' id='mathml_44f470209ecd7206b094a1bff11a171ba68ae7a1_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> satisfying the countable chain condition is Lindelöf. \end{theorem}</p> <p>\begin{proof} The proof goes by contradiction: Assume there is an open cover <math class='maruku-mathml' display='inline' id='mathml_44f470209ecd7206b094a1bff11a171ba68ae7a1_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>U</mi> <mi>i</mi></msub><msub><mo stretchy='false'>}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>\{U_i\}_{i\in I}</annotation></semantics></math> with no countable subcover. Let <math class='maruku-mathml' display='inline' id='mathml_44f470209ecd7206b094a1bff11a171ba68ae7a1_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>U</mi> <mi>j</mi></msub><msub><mo stretchy='false'>}</mo> <mrow><mi>j</mi><mo>∈</mo><mi>J</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>\{U_j\}_{j\in J}</annotation></semantics></math> be a locally finite refinement, which again must not be countable. This is to say that each point possesses an open neighborhood so small that it is only an element of finitely many <math class='maruku-mathml' display='inline' id='mathml_44f470209ecd7206b094a1bff11a171ba68ae7a1_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>U</mi> <mi>j</mi></msub></mrow><annotation encoding='application/x-tex'>U_j</annotation></semantics></math>‘s. Inductively (using <a class='existingWikiWord' href='/nlab/show/diff/Zorn%27s+lemma'>Zorn's lemma</a>) construct a maximal system <math class='maruku-mathml' display='inline' id='mathml_44f470209ecd7206b094a1bff11a171ba68ae7a1_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>V</mi> <mi>λ</mi></msub><msub><mo stretchy='false'>}</mo> <mrow><mi>λ</mi><mo>∈</mo><mi>Λ</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>\{V_\lambda\}_{\lambda \in \Lambda}</annotation></semantics></math> of pairwise disjoint opens being contained in at most finitely many <math class='maruku-mathml' display='inline' id='mathml_44f470209ecd7206b094a1bff11a171ba68ae7a1_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>U</mi> <mi>j</mi></msub></mrow><annotation encoding='application/x-tex'>U_j</annotation></semantics></math>’s. Due to maximality <math class='maruku-mathml' display='inline' id='mathml_44f470209ecd7206b094a1bff11a171ba68ae7a1_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mo lspace='thinmathspace' rspace='thinmathspace'>⋃</mo> <mrow><mi>λ</mi><mo>∈</mo><mi>Λ</mi></mrow></msub><msub><mi>V</mi> <mi>λ</mi></msub></mrow><annotation encoding='application/x-tex'>\bigcup_{\lambda\in\Lambda} V_\lambda</annotation></semantics></math> is dense. This fact implies by the countable chain condition that <math class='maruku-mathml' display='inline' id='mathml_44f470209ecd7206b094a1bff11a171ba68ae7a1_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Λ</mi></mrow><annotation encoding='application/x-tex'>\Lambda</annotation></semantics></math> is countable. Moreover it implies that each <math class='maruku-mathml' display='inline' id='mathml_44f470209ecd7206b094a1bff11a171ba68ae7a1_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>U</mi> <mi>j</mi></msub></mrow><annotation encoding='application/x-tex'>U_j</annotation></semantics></math> intersects at least one <math class='maruku-mathml' display='inline' id='mathml_44f470209ecd7206b094a1bff11a171ba68ae7a1_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>V</mi> <mi>λ</mi></msub></mrow><annotation encoding='application/x-tex'>V_\lambda</annotation></semantics></math>. But this is to say that there are at most countably many <math class='maruku-mathml' display='inline' id='mathml_44f470209ecd7206b094a1bff11a171ba68ae7a1_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>U</mi> <mi>j</mi></msub></mrow><annotation encoding='application/x-tex'>U_j</annotation></semantics></math>’s. This is a contradiction. \end{proof}</p> <h2 id='related_statements_and_properties'>Related statements and properties</h2> <p><em>Could not include topology - global countability axioms</em></p> </div> <div class="revisedby"> <p> Last revised on April 3, 2020 at 19:37:40. 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