The characterizations of hyperspaces and free topological groups with an $ω^ω$-base
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866910847669895168 |
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| author | Lin, Fucai Liu, Chuan |
| author_facet | Lin, Fucai Liu, Chuan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_19727 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The characterizations of hyperspaces and free topological groups with an $ω^ω$-base Lin, Fucai Liu, Chuan General Topology Primary 54B20, secondary 22A05, 5430, 54D45, 54D70, 54E35 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. |
| title | The characterizations of hyperspaces and free topological groups with an $ω^ω$-base |
| topic | General Topology Primary 54B20, secondary 22A05, 5430, 54D45, 54D70, 54E35 |
| url | https://arxiv.org/abs/2502.19727 |