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On topological groups with a small base and metrizability

100%

Gabriyelyan S., Kąkol J., Leiderman A.

Fundamenta Mathematicae

2015

tom 229

nr 2

129-158

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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On inductive limits of topological algebras

62%

Kąkol J.

Colloquium Mathematicum

1982

tom 47

nr 1

71-78

Ograniczanie wyników

1 Colloquium Mathematicum

1 Fundamenta Mathematicae

2 Kąkol J.

1 Gabriyelyan S.

1 Leiderman A.

1 2015

1 1982

Wyniki wyszukiwania

On topological groups with a small base and metrizability

On inductive limits of topological algebras