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

Abstract

A topological space $(X, \tau)$ is said to be have an {\it $\omega^\omega$-base} if for each point $x\in X$ there exists a neighborhood base $\{U_{\alpha}[x]: \alpha\in\omega^\omega\}$ such that $U_{\beta}[x]\subset U_{\alpha}[x]$ for all $\alpha\leq\beta$ in $\omega^\omega$. 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 $\omega^\omega$-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 $\omega^\omega$-base if and only if $X$ is a topological sum of a discrete space and a submetrizable $k_\omega$-space. (2) If $X$ is a metrizable space, then $(CL(X), \tau_V)$ has an $\omega^\omega$-base if and only if $X$ is separable and the boundary of each closed subset of $X$ is $\sigma$-compact. (3) If $X$ is a metrizable space, then $(CL(X), \tau_F)$ has an $\omega^\omega$-base consisting of basic neighborhoods if and only if $X$ is a Polish space. (4) If $X$ is a metrizable space, then $(CL(X), \tau_F)$ is a Fr\'echet-Urysohn space with an $\omega^\omega$-base, if and only if $(CL(X), \tau_F)$ is first-countable, if and only if $X$ is a locally compact and second countable space.