Modulo arithmetic of function spaces: Subset hyperspaces as quotients of function spaces
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866909907026968576 |
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| author | Akofor, Earnest |
| author_facet | Akofor, Earnest |
| contents | Let $X$ be a (topological) space and $Cl(X)$ the collection of nonempty closed subsets of $X$. Given a topology on $Cl(X)$, making $Cl(X)$ a space, a (subset) hyperspace of $X$ is a subspace $\mathcal{J}\subset Cl(X)$ with an embedding $X\hookrightarrow\mathcal{J}$, $x\mapsto\{x\}$. In this note, we characterize certain hyperspaces $\mathcal{J}\subset Cl(X)$ as explicit quotient spaces of function spaces $\mathcal{F}\subset X^Y$ and discuss metrization of associated compact-subset hyperspaces in this setting. In particular, we find that any hyperspace topology containing the Vietoris topology is a quotient of a function space topology containing the topology of pointwise convergence. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_10803 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Modulo arithmetic of function spaces: Subset hyperspaces as quotients of function spaces Akofor, Earnest General Topology Primary 54B20 54C35, Secondary 54B15 54E35 54A10 54C50 Let $X$ be a (topological) space and $Cl(X)$ the collection of nonempty closed subsets of $X$. Given a topology on $Cl(X)$, making $Cl(X)$ a space, a (subset) hyperspace of $X$ is a subspace $\mathcal{J}\subset Cl(X)$ with an embedding $X\hookrightarrow\mathcal{J}$, $x\mapsto\{x\}$. In this note, we characterize certain hyperspaces $\mathcal{J}\subset Cl(X)$ as explicit quotient spaces of function spaces $\mathcal{F}\subset X^Y$ and discuss metrization of associated compact-subset hyperspaces in this setting. In particular, we find that any hyperspace topology containing the Vietoris topology is a quotient of a function space topology containing the topology of pointwise convergence. |
| title | Modulo arithmetic of function spaces: Subset hyperspaces as quotients of function spaces |
| topic | General Topology Primary 54B20 54C35, Secondary 54B15 54E35 54A10 54C50 |
| url | https://arxiv.org/abs/2503.10803 |