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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2506.18733 |
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Table of 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.