On topological groups with a small base and metrizability
Volume 229 / 2015
Abstract
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.
