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| Format: | Preprint |
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2025
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| Accès en ligne: | https://arxiv.org/abs/2509.05105 |
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| _version_ | 1866918296245239808 |
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| author | Lipin, Anton Reznichenko, Evgenii Sipacheva, Ol'ga |
| author_facet | Lipin, Anton Reznichenko, Evgenii Sipacheva, Ol'ga |
| contents | A topological space $X$ is $\mathbb R^{ω_1}$-factorizable if any continuous function $f\colon X\to \mathbb R^{ω_1}$ factors through a continuous function from $X$ to a second-countable space. It is shown that a Tychonoff space $X$ is $\mathbb R^{ω_1}$-factorizable if and only if $X\times D(ω_1)$, where $D(ω_1)$ is a discrete space of cardinality $ω_1$, is $z$-embedded in the product $βX\times βD(ω_1)$ of the Stone--Cech compactifications. It is also proved that $\mathbb R^{ω_1}$-factorizability is hereditary and countably multiplicative, that any $\mathbb R^{ω_1}$-factorizable space is hereditarily Lindelöf and hereditarily separable, and that the existence of nonmetrizable $\mathbb R^{ω_1}$-factorizable topological spaces and groups is independent of ZFC: under CH, all $\mathbb R^{ω_1}$-factorizable spaces are second-countable, while under MA + $\lnot$CH, the countable Fréchet--Urysohn fan is $\mathbb R^{ω_1}$-factorizable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_05105 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $\mathbb R^{ω_1}$-Factorizable Spaces and Groups Lipin, Anton Reznichenko, Evgenii Sipacheva, Ol'ga General Topology 22A05, 54F45 A topological space $X$ is $\mathbb R^{ω_1}$-factorizable if any continuous function $f\colon X\to \mathbb R^{ω_1}$ factors through a continuous function from $X$ to a second-countable space. It is shown that a Tychonoff space $X$ is $\mathbb R^{ω_1}$-factorizable if and only if $X\times D(ω_1)$, where $D(ω_1)$ is a discrete space of cardinality $ω_1$, is $z$-embedded in the product $βX\times βD(ω_1)$ of the Stone--Cech compactifications. It is also proved that $\mathbb R^{ω_1}$-factorizability is hereditary and countably multiplicative, that any $\mathbb R^{ω_1}$-factorizable space is hereditarily Lindelöf and hereditarily separable, and that the existence of nonmetrizable $\mathbb R^{ω_1}$-factorizable topological spaces and groups is independent of ZFC: under CH, all $\mathbb R^{ω_1}$-factorizable spaces are second-countable, while under MA + $\lnot$CH, the countable Fréchet--Urysohn fan is $\mathbb R^{ω_1}$-factorizable. |
| title | $\mathbb R^{ω_1}$-Factorizable Spaces and Groups |
| topic | General Topology 22A05, 54F45 |
| url | https://arxiv.org/abs/2509.05105 |