Compactly generated group
In
mathematics, a
compactly generated (topological) group is a
topological group G which is
algebraically generated by one of its
compact subsets. Explicitly, this means that there exists a compact subset
K of
G such that
- .
So if
K is symmetric, i.e.
K =
K −1, then
- .
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
Un 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).