On topological groups with a small base and metrizability

Saak Gabriyelyan¹ ; Jerzy Kąkol² ; Arkady Leiderman¹

¹ Department of Mathematics Ben-Gurion University of the Negev Beer Sheva, Israel
² Faculty of Mathematics and Informatics A. Mickiewicz University 61-614 Poznań, Poland

Fundamenta Mathematicae, Tome 229 (2015) no. 2, pp. 129-158.

Résumé

A (Hausdorff) topological group is said to have a $\mathfrak {G}$-base if it admits a base of neighbourhoods of the unit, $\{U_{\alpha }: \alpha \in \mathbb {N}^{\mathbb {N}}\}$, such that $U_{\alpha }\subset U_{\beta }$ whenever $\beta \leq \alpha $ for all $\alpha ,\beta \in \mathbb {N}^{\mathbb {N}}$. The class of all metrizable topological groups is a proper subclass of the class $\mathbf {TG}_\mathfrak {G}$ of all topological groups having a $\mathfrak {G}$-base. We prove that a topological group is metrizable iff it is Fréchet–Urysohn and has a $\mathfrak {G}$-base. We also show that any precompact set in a topological group $G\in \mathbf {TG}_\mathfrak {G}$ is metrizable, and hence $G$ is strictly angelic. We deduce from this result that an almost metrizable group is metrizable iff it has a $\mathfrak {G}$-base. Characterizations of metrizability of topological vector spaces, in particular of $C_{c}(X)$, are given using $\mathfrak {G}$-bases. We prove that if $X$ is a submetrizable $k_\omega $-space, then the free abelian topological group $A(X)$ and the free locally convex topological space $L(X)$ have a $\mathfrak {G}$-base. Another class $\mathbf {TG}_\mathcal {CR}$ of topological groups with a compact resolution swallowing compact sets appears naturally. We show that $\mathbf {TG}_\mathcal {CR}$ and $\mathbf {TG}_\mathfrak {G}$ are in some sense dual to each other. We conclude with a dozen open questions and various (counter)examples.

Détail

DOI : 10.4064/fm229-2-3

Keywords: hausdorff topological group said have mathfrak base admits base neighbourhoods unit alpha alpha mathbb mathbb alpha subset beta whenever beta leq alpha alpha beta mathbb mathbb class metrizable topological groups proper subclass class mathbf mathfrak topological groups having mathfrak base prove topological group metrizable chet urysohn has mathfrak base precompact set topological group mathbf mathfrak metrizable hence strictly angelic deduce result almost metrizable group metrizable has mathfrak base characterizations metrizability topological vector spaces particular given using mathfrak bases prove submetrizable omega space abelian topological group locally convex topological space have mathfrak base another class mathbf mathcal topological groups compact resolution swallowing compact sets appears naturally mathbf mathcal mathbf mathfrak sense dual each other conclude dozen questions various counter examples