imported>David Eppstein: does not need more categories

2025-04-22T00:45:53Z

does not need more categories

New page

{{Multiple issues|{{more sources|date=March 2025}}
{{one source|date=March 2025}}}}

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.<ref>{{citation|title=Locally Compact Groups|first=Markus|last=Stroppel|publisher=European Mathematical Society|year=2006|isbn=9783037190166|page=44|url=https://books.google.com/books?id=3_BPupMDRr8C&pg=PA44}}.</ref> 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.

== Definition ==

A [[topological group]] ''G'' is said to be '''compactly generated''' if there exists a compact subset ''K'' of ''G'' such that

:<math>\langle K\rangle = \bigcup_{n \in \mathbb{N}} (K \cup K^{-1})^n = G.</math>

So if ''K'' is symmetric, i.e. ''K'' = ''K''<sup> −1</sup>, then

:<math>G = \bigcup_{n \in \mathbb{N}} K^n.</math>

== Locally compact case ==

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

:''U''<sub>''n''</sub>

of open identity neighborhoods, there exists a [[normal subgroup]] ''N'' contained in the intersection of that sequence, such that

:''G''/''N''

is locally compact metric separable (the [[Kakutani-Kodaira-Montgomery-Zippin theorem]]).

==References==
{{reflist}}

[[Category:Topological groups]]
[[Category:Topology]]
{{topology-stub}}

Compactly generated group - Revision history

imported>David Eppstein: does not need more categories