Weird \mathbb R-Factorizable Groups

2025/06/23 by Evgenii Reznichenko, Reznichenko, Evgenii, Ol'ga Sipacheva +1 · 1 voice
Mathematics · #22A05 #54F45 #FOS: Mathematics #General Topology (math.GN) #math.GN

paper · pdf · doi:10.48550/arxiv.2506.18733

Abstract

The problem of the existence of non-pseudo-ℵ₁-compact \mathbb R-factorizable groups is studied. It is proved that any such group is submetrizable and has weight larger than ω₁. Closely related results concerning the \mathbb R-factorizability of products of topological groups and spaces are also obtained (a product X× Y of topological spaces is said to be \mathbb R-factorizable if any continuous function X× Y→ \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× 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-ℵ₁-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.

Citations

• Extensions and factorizations of topological and semitopological universal algebras
• No Subgroup Theorem for the Covering Dimension of Topological Groups
• No Product Theorem for the Covering Dimension of Topological Groups

Discussions

• Evgenii Reznichenko, Ol'ga Sipacheva: Weird $\mathbb R$-Factorizable Groups https://arxiv.org/abs/2506.18733 https://arxiv.org/pdf/2506.18733 https://arxiv.org/html/2506.18733 [bsky, 0 points, 0 comments]

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