On uniformly continuous surjections between $C_p$-spaces over metrizable spaces
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910926427389952 |
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| author | Eysen, A. Leiderman, A. Valov, V. |
| author_facet | Eysen, A. Leiderman, A. Valov, V. |
| contents | Let $X$ be metrizable, $Y$ be perfectly normal and suppose that there exists a uniformly continuous surjection $T: C_{p}(X) \to C_{p}(Y)$ (resp., $T: C_{p}^*(X) \to C_{p}^*(Y)$), where $C_{p}(X)$ (resp., $C_{p}^*(X)$) denotes the space of all real-valued continuous (resp., continuous and bounded) functions on $X$ endowed with the pointwise convergence topology.
We show that if additionally $T$ is an inversely bounded mapping and $X$ has some dimensional-like property $\mathcal P$, then so does $Y$. For example, this is true if $\mathcal P$ is one of the following properties: zero-dimensionality, countable-dimensionality or strong countable-dimensionality.
Also, we consider other properties $\mathcal P$: of being a scattered, or a strongly $σ$-scattered space, or being a $Δ_1$-space (see [17]). Our results strengthen and extend several results from [6], [13], [17]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_01870 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On uniformly continuous surjections between $C_p$-spaces over metrizable spaces Eysen, A. Leiderman, A. Valov, V. General Topology 54C35, 54F45 Let $X$ be metrizable, $Y$ be perfectly normal and suppose that there exists a uniformly continuous surjection $T: C_{p}(X) \to C_{p}(Y)$ (resp., $T: C_{p}^*(X) \to C_{p}^*(Y)$), where $C_{p}(X)$ (resp., $C_{p}^*(X)$) denotes the space of all real-valued continuous (resp., continuous and bounded) functions on $X$ endowed with the pointwise convergence topology. We show that if additionally $T$ is an inversely bounded mapping and $X$ has some dimensional-like property $\mathcal P$, then so does $Y$. For example, this is true if $\mathcal P$ is one of the following properties: zero-dimensionality, countable-dimensionality or strong countable-dimensionality. Also, we consider other properties $\mathcal P$: of being a scattered, or a strongly $σ$-scattered space, or being a $Δ_1$-space (see [17]). Our results strengthen and extend several results from [6], [13], [17]. |
| title | On uniformly continuous surjections between $C_p$-spaces over metrizable spaces |
| topic | General Topology 54C35, 54F45 |
| url | https://arxiv.org/abs/2408.01870 |