Topological groups with dense compactly generated subgroups
Submitted: 2014-01-27
|Accepted: 2014-01-27
|Published: 2002-04-01
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Keywords:
Topological group, Compactly generated group, Dense subgroup, Almost metrizable group, ℵ0-bounded group, Paracompact p-space, Metric space, σ-compact space, Space of countable type
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Abstract:
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.
References:
H. Fujita and D. B. Shakhmatov, A characterization of compactly generated metrizable groups, Proc. Amer. Math. Soc., to appear.
I. Guran, Topological groups similar to Lindelöf groups (in Russian), Dokl. Akad. Nauk SSSR 256 (1981), no. 6, 1305-1307; English translation in: Soviet Math. Dokl. 23 (1981), no. 1, 173-175.
B.A. Pasynkov, Almost-metrizable topological groups (in Russian), Dokl. Akad. Nauk SSSR 161 (1965), 281-284.
