Compactly Generated Group Article Index for
Compactly
Website Links For
Group
 

Information About

Compactly Generated Group




:\langle K angle = \bigcup_{n \in \mathbb{N}} (K^n \cup K^{-n}) = G.

So if ''K'' is symmetric, i.e. ''K'' = ''K'' −1, then

:G = \bigcup_{n \in \mathbb{N}} K^n.

This property is interesting in the case of Locally Compact topological groups, since locally compact compactly generated topological groups can be approximated by locally compact, Separable Metric factor groups of ''G'', in the sense that for a sequence ''U''''n'' 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 ).