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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Compactly generated group</span></span>
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<p>In <a href="Mathematics" title="Mathematics">mathematics</a>, a <b>compactly generated (topological) group</b> is a <a href="Topological_group" title="Topological group">topological group</a> <i>G</i> which is <a href="Generating_set_of_a_group" title="Generating set of a group">algebraically generated</a> by one of its <a href="Compact_space" title="Compact space">compact</a> subsets.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> This should not be confused with the unrelated notion (widely used in <a href="Algebraic_topology" title="Algebraic topology">algebraic topology</a>) of a <a href="Compactly_generated_space" title="Compactly generated space">compactly generated space</a> -- one whose <a href="Topology" title="Topology">topology</a> is generated (in a suitable sense) by its compact subspaces.
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<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>
<p>A <a href="Topological_group" title="Topological group">topological group</a> <i>G</i> is said to be <b>compactly generated</b> if there exists a compact subset <i>K</i> of <i>G</i> such that
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle K\rangle =\bigcup _{n\in \mathbb {N} }(K\cup K^{-1})^{n}=G.}">
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<annotation encoding="application/x-tex">{\displaystyle \langle K\rangle =\bigcup _{n\in \mathbb {N} }(K\cup K^{-1})^{n}=G.}</annotation>
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</math></span><img src="./4bd7ad0d149a392c1e5aecea996db8f8fddb2f2f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:27.918ex; height:5.676ex;" alt="{\displaystyle \langle K\rangle =\bigcup _{n\in \mathbb {N} }(K\cup K^{-1})^{n}=G.}" loading="lazy"></span></dd></dl>
<p>So if <i>K</i> is symmetric, i.e. <i>K</i> = <i>K</i><sup> −1</sup>, then
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G=\bigcup _{n\in \mathbb {N} }K^{n}.}">
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<annotation encoding="application/x-tex">{\displaystyle G=\bigcup _{n\in \mathbb {N} }K^{n}.}</annotation>
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<div class="mw-heading mw-heading2"><h2 id="Locally_compact_case">Locally compact case</h2></div>
<p>This property is interesting in the case of <a href="Locally_compact_space" title="Locally compact space">locally compact</a> topological groups, since locally compact compactly generated topological groups can be approximated by locally compact, <a href="Separable_space" title="Separable space">separable</a> <a href="Metric_space" title="Metric space">metric</a> factor groups of <i>G</i>. More precisely, for a sequence
</p>
<dl><dd><i>U</i><sub><i>n</i></sub></dd></dl>
<p>of open identity neighborhoods, there exists a <a href="Normal_subgroup" title="Normal subgroup">normal subgroup</a> <i>N</i> contained in the intersection of that sequence, such that
</p>
<dl><dd><i>G</i>/<i>N</i></dd></dl>
<p>is locally compact metric separable (the Kakutani-Kodaira-Montgomery-Zippin theorem).
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<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
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</style><cite id="CITEREFStroppel2006" class="citation cs2">Stroppel, Markus (2006), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=3_BPupMDRr8C&pg=PA44"><i>Locally Compact Groups</i></a>, European Mathematical Society, p. 44, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9783037190166</bdi></cite>.</span>
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