The characterizations of hyperspaces and free topological groups with an $ω^ω$-base

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.