A topological groupG is said to be compactly generated if there exists a compact subset K of G such that
So if K is symmetric, i.e. K = K −1, then
Locally compact caseedit
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, separablemetric factor groups of G. More precisely, for a sequence
Un
of open identity neighborhoods, there exists a normal subgroupN contained in the intersection of that sequence, such that
G/N
is locally compact metric separable (the Kakutani-Kodaira-Montgomery-Zippin theorem).
Referencesedit
^Stroppel, Markus (2006), Locally Compact Groups, European Mathematical Society, p. 44, ISBN9783037190166.
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compactly, generated, group, mathematics, compactly, generated, topological, group, topological, group, which, algebraically, generated, compact, subsets, this, should, confused, with, unrelated, notion, widely, used, algebraic, topology, compactly, generated,. In mathematics a compactly generated topological group is a topological group G which is algebraically generated by one of its compact subsets 1 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 editA topological group G is said to be compactly generated if there exists a compact subset K of G such that K n N K K 1 n G displaystyle langle K rangle bigcup n in mathbb N K cup K 1 n G nbsp So if K is symmetric i e K K 1 then G n NKn displaystyle G bigcup n in mathbb N K n nbsp Locally compact case editThis 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 More precisely for a sequence Unof open identity neighborhoods there exists a normal subgroup N contained in the intersection of that sequence such that G Nis locally compact metric separable the Kakutani Kodaira Montgomery Zippin theorem References edit Stroppel Markus 2006 Locally Compact Groups European Mathematical Society p 44 ISBN 9783037190166 nbsp This topology related article is a stub You can help Wikipedia by expanding it vte Retrieved from https en wikipedia org w index php title Compactly generated group amp oldid 743601072, wikipedia, wiki, book, books, library,