The characterizations of hyperspaces and free topological groups with an $ω^ω$-base
Fucai Lin, Chuan Liu
TL;DR
The paper characterizes when key topological constructions admit an $ω^ω$-base. It shows $A(X)$ has an $ω^ω$-base (as a $k$-space) exactly when $X$ is a topological sum of a discrete space and a submetrizable $k_ω$-space, extending prior results on free topological groups. For metrizable $X$, it characterizes $(CL(X),τ_V)$ as $ω^ω$-base exactly when $X$ is separable and the boundary of every closed set is σ-compact, and $(CL(X),τ_F)$ as $ω^ω$-base of basic neighborhoods exactly when $X$ is Polish; it also links (F)réchet–Urysohn properties to first-countability and to local compactness with second countability. Collectively, these results delineate sharp structural boundaries for hyperspaces and free Abelian topological groups with $ω^ω$-bases, with implications for separable Polish spaces and locally compact second-countable spaces.
Abstract
A topological space $(X, τ)$ is said to be have an {\it $ω^ω$-base} if for each point $x\in X$ there exists a neighborhood base $\{U_α[x]: α\inω^ω\}$ such that $U_β[x]\subset U_α[x]$ for all $α\leqβ$ in $ω^ω$. In this paper, the characterization of a space $X$ is given such that the free Abelian topological group $A(X)$, the hyperspace $CL(X)$ with the Vietoris topology and the hyperspace $CL(X)$ with the Fell topology have $ω^ω$-bases respectively. The main results are listed as follows: (1) For a Tychonoff space $X$, the free Abelian topological group $A(X)$ is a $k$-space with an $ω^ω$-base if and only if $X$ is a topological sum of a discrete space and a submetrizable $k_ω$-space. (2) If $X$ is a metrizable space, then $(CL(X), τ_V)$ has an $ω^ω$-base if and only if $X$ is separable and the boundary of each closed subset of $X$ is $σ$-compact. (3) If $X$ is a metrizable space, then $(CL(X), τ_F)$ has an $ω^ω$-base consisting of basic neighborhoods if and only if $X$ is a Polish space. (4) If $X$ is a metrizable space, then $(CL(X), τ_F)$ is a Fréchet-Urysohn space with an $ω^ω$-base, if and only if $(CL(X), τ_F)$ is first-countable, if and only if $X$ is a locally compact and second countable space.