$\mathbb R^{\omega_1}$-Factorizable Spaces and Groups

Abstract

A topological space $X$ is $\mathbb R^{\omega_1}$-factorizable if any continuous function $f\colon X\to \mathbb R^{\omega_1}$ factors through a continuous function from $X$ to a second-countable space. It is shown that a Tychonoff space $X$ is $\mathbb R^{\omega_1}$-factorizable if and only if $X\times D(\omega_1)$, where $D(\omega_1)$ is a discrete space of cardinality $\omega_1$, is $z$-embedded in the product $\beta X\times \beta D(\omega_1)$ of the Stone--Cech compactifications. It is also proved that $\mathbb R^{\omega_1}$-factorizability is hereditary and countably multiplicative, that any $\mathbb R^{\omega_1}$-factorizable space is hereditarily Lindel\"of and hereditarily separable, and that the existence of nonmetrizable $\mathbb R^{\omega_1}$-factorizable topological spaces and groups is independent of ZFC: under CH, all $\mathbb R^{\omega_1}$-factorizable spaces are second-countable, while under MA + $\lnot$CH, the countable Fr\'echet--Urysohn fan is $\mathbb R^{\omega_1}$-factorizable.