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 $\alpha \le \beta$ 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 Fréchet-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.

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