Weird $\mathbb R$-Factorizable Groups

TL;DR

This paper investigates the properties of $ ext{R}$-factorizable topological groups, establishing conditions under which such groups are submetrizable, have large weight, and how their product spaces behave in terms of factorization.

Contribution

It proves that non-pseudo-$ ext{aleph}_1$-compact $ ext{R}$-factorizable groups are submetrizable with large weight and characterizes when products of groups and spaces are $ ext{R}$-factorizable, linking to properties like hereditarily separable and Lindelöf.

Findings

Non-pseudo-$ ext{aleph}_1$-compact $ ext{R}$-factorizable groups are submetrizable.

The $ ext{R}$-factorizability of a product $X imes Y$ depends on factorization properties of $X$ and $Y$.

If $X imes$ uncountable discrete space is $ ext{R}$-factorizable, then $X^ ext{omega}$ is hereditarily separable and Lindelöf.

Abstract

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 $\omega_1$. 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 $X^\omega$ is heredirarily separable and heredirarily Lindel\"of.