Topological groups with dense compactly generated subgroups
Keywords:Topological group, Compactly generated group, Dense subgroup, Almost metrizable group, â„µ0-bounded group, Paracompact p-space, Metric space, σ-compact space, Space of countable type
A topological group G is: (i) compactly generated if it contains a compact subset algebraically generating G, (ii) -compact if G is a union of countably many compact subsets, (iii) 0-bounded if arbitrary neighborhood U of the identity element of G has countably many translates xU that cover G, and (iv) finitely generated modulo open sets if for every non-empty open subset U of G there exists a finite set F such that F U algebraically generates G. We prove that: (1) a topological group containing a dense compactly generated subgroup is both 0-bounded and finitely generated modulo open sets, (2) an almost metrizable topological group has a dense compactly generated subgroup if and only if it is both 0-bounded and finitely generated modulo open sets, and (3) an almost metrizable topological group is compactly generated if and only if it is -compact and finitely generated modulo open sets.
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B.A. Pasynkov, Almost-metrizable topological groups (in Russian), Dokl. Akad. Nauk SSSR 161 (1965), 281-284.
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