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| {{Unreferenced|date=April 2008}}
| | Eusebio Stanfill is what's blogged on my birth certificate although it is always the name on my birth certificate. Idaho is our birth internet site. I work as an order clerk. As a man what When i really like is representing but I'm thinking on starting something new. You is likely to find my website here: http://circuspartypanama.com<br><br>my website [http://circuspartypanama.com clash of clans unlimited troops] |
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| In [[mathematics]], a '''compactly generated (topological) group''' is a [[topological group]] ''G'' which is [[generating set of a group|algebraically generated]] by one of its [[compact space|compact]] subsets. This should not be confused with the unrelated notion (widely used in [[algebraic topology]]) of a [[compactly generated space]] -- one whose [[topology]] is generated (in a suitable sense) by its compact subspaces.
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| == Definition ==
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| A [[topological group]] ''G'' is said to be '''compactly generated''' if there exists a compact subset ''K'' of ''G'' such that
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| :<math>\langle K\rangle = \bigcup_{n \in \mathbb{N}} (K \cup K^{-1})^n = G.</math> | |
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| So if ''K'' is symmetric, i.e. ''K'' = ''K''<sup> −1</sup>, then
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| :<math>G = \bigcup_{n \in \mathbb{N}} K^n.</math>
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| == Locally compact case ==
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| This property is interesting in the case of [[Locally compact space|locally compact]] topological groups, since locally compact compactly generated topological groups can be approximated by locally compact, [[separable space|separable]] [[metric space|metric]] factor groups of ''G''. More precisely, for a sequence
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| :''U''<sub>''n''</sub> | |
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| of open identity neighborhoods, there exists a [[normal subgroup]] ''N'' contained in the intersection of that sequence, such that
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| :''G''/''N''
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| is locally compact metric separable (the [[Kakutani-Kodaira-Montgomery-Zippin theorem]]).
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| [[Category:Topological groups]]
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| {{topology-stub}}
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Latest revision as of 05:04, 29 December 2014
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