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| Natura: | Preprint |
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2025
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| Accesso online: | https://arxiv.org/abs/2506.18733 |
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| _version_ | 1866908417339162624 |
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| author | Reznichenko, Evgenii Sipacheva, Ol'ga |
| author_facet | Reznichenko, Evgenii Sipacheva, Ol'ga |
| contents | The problem of the existence of non-pseudo-$\aleph_1$-compact $\mathbb R$-factorizable groups is studied. It is proved that any such group is submetrizable and has weight larger than $ω_1$. Closely related results concerning the $\mathbb R$-factorizability of products of topological groups and spaces are also obtained (a product $X\times Y$ of topological spaces is said to be $\mathbb R$-factorizable if any continuous function $X\times Y\to \mathbb R$ factors through a product of maps from $X$ and $Y$ to second-countable spaces). In particular, it is proved that the square $G\times G$ of a topological groups $G$ is $\mathbb R$-factorizable as a group if and only if it is $\mathbb R$-factorizable as a product of spaces, in which case $G$ is pseudo-$\aleph_1$-compact. It is also proved that if the product of a space $X$ and an uncountable discrete space is $\mathbb R$-factorizable, then $X^ω$ is heredirarily separable and heredirarily Lindelöf. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_18733 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Weird $\mathbb R$-Factorizable Groups Reznichenko, Evgenii Sipacheva, Ol'ga General Topology 22A05, 54F45 The problem of the existence of non-pseudo-$\aleph_1$-compact $\mathbb R$-factorizable groups is studied. It is proved that any such group is submetrizable and has weight larger than $ω_1$. Closely related results concerning the $\mathbb R$-factorizability of products of topological groups and spaces are also obtained (a product $X\times Y$ of topological spaces is said to be $\mathbb R$-factorizable if any continuous function $X\times Y\to \mathbb R$ factors through a product of maps from $X$ and $Y$ to second-countable spaces). In particular, it is proved that the square $G\times G$ of a topological groups $G$ is $\mathbb R$-factorizable as a group if and only if it is $\mathbb R$-factorizable as a product of spaces, in which case $G$ is pseudo-$\aleph_1$-compact. It is also proved that if the product of a space $X$ and an uncountable discrete space is $\mathbb R$-factorizable, then $X^ω$ is heredirarily separable and heredirarily Lindelöf. |
| title | Weird $\mathbb R$-Factorizable Groups |
| topic | General Topology 22A05, 54F45 |
| url | https://arxiv.org/abs/2506.18733 |