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## Fundamenta Mathematicae

2015 | 229 | 2 | 129-158
Tytuł artykułu

### On topological groups with a small base and metrizability

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EN
A (Hausdorff) topological group is said to have a 𝔊-base if it admits a base of neighbourhoods of the unit, ${U_{α}: α ∈ ℕ^{ℕ}}$, such that $U_{α} ⊂ U_{β}$ whenever β ≤ α for all $α, β ∈ ℕ^{ℕ}$. The class of all metrizable topological groups is a proper subclass of the class $TG_{𝔊}$ of all topological groups having a 𝔊-base. We prove that a topological group is metrizable iff it is Fréchet-Urysohn and has a 𝔊-base. We also show that any precompact set in a topological group $G ∈ TG_{𝔊}$ is metrizable, and hence G is strictly angelic. We deduce from this result that an almost metrizable group is metrizable iff it has a 𝔊-base. Characterizations of metrizability of topological vector spaces, in particular of $C_{c}(X)$, are given using 𝔊-bases. We prove that if X is a submetrizable $k_{ω}$-space, then the free abelian topological group A(X) and the free locally convex topological space L(X) have a 𝔊-base. Another class $TG_{𝒞𝓡}$ of topological groups with a compact resolution swallowing compact sets appears naturally. We show that $TG_{𝒞𝓡}$ and $TG_{𝔊}$ are in some sense dual to each other. We conclude with a dozen open questions and various (counter)examples.
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Tom
Numer
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129-158
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wydano
2015
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autor
• Department of Mathematics, Ben-Gurion University of the Negev, Beer Sheva, Israel
autor
• Faculty of Mathematics and Informatics, A. Mickiewicz University, 61-614 Poznań, Poland
autor
• Department of Mathematics, Ben-Gurion University of the Negev, Beer Sheva, Israel
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