On topological groups with a small base and metrizability

• Autorzy: Saak Gabriyelyan , Jerzy Kąkol , Arkady Leiderman
• Języki publikacji: EN

Abstrakty

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.

Kategorie tematyczne

• 54C35: Function spaces
• 22A05: Structure of general topological groups
• 54H11: Topological groups
• 46A50: Compactness in topological linear spaces; angelic spaces, etc.

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2015
• Tom: 229
• Numer: 2
• Strony: 129-158

Daty

wydano

2015

Twórcy

autor

Saak Gabriyelyan

• Department of Mathematics, Ben-Gurion University of the Negev, Beer Sheva, Israel

autor

Jerzy Kąkol

• Faculty of Mathematics and Informatics, A. Mickiewicz University, 61-614 Poznań, Poland

autor

Arkady Leiderman

• Department of Mathematics, Ben-Gurion University of the Negev, Beer Sheva, Israel

Identyfikatory

DOI

10.4064/fm229-2-3