$\mathbb R^{ω_1}$-Factorizable Spaces and Groups
Anton Lipin, Evgenii Reznichenko, Ol'ga Sipacheva
TL;DR
The paper investigates spaces $X$ for which $X\times D(\omega_1)$ is $\mathbb R$-factorizable, by introducing and exploiting $\mathbb R^{\omega_1}$-factorizability and its intimate link with $A$-filters. It develops multiple equivalent characterizations, establishes stability under countable products and $\omega$-powers, and shows that neighborhood structures must form $A$-filters, tying local filter properties to global factorizability. The work yields hereditary, countably multiplicative behavior and hereditary Lindelöf/separability for $\mathbb R^{\omega_1}$-factorizable spaces, and demonstrates independence results: under CH all such spaces are second-countable, while under MA+$\lnot$CH there exist nonmetrizable examples, notably via the Fréchet–Urysohn fan. It further extends these ideas to topological groups, giving a characterization in terms of factorization through second-countable groups and introducing the notions of $\omega$-narrowness and $\omega_1$-fineness, with connections to $A$-filters and open questions on product behavior and filters. Overall, the paper clarifies the set-theoretic boundaries of $\mathbb R^{\omega_1}$-factorizability and supplies tools for analyzing factorization in spaces and groups.
Abstract
A topological space $X$ is $\mathbb R^{ω_1}$-factorizable if any continuous function $f\colon X\to \mathbb R^{ω_1}$ factors through a continuous function from $X$ to a second-countable space. It is shown that a Tychonoff space $X$ is $\mathbb R^{ω_1}$-factorizable if and only if $X\times D(ω_1)$, where $D(ω_1)$ is a discrete space of cardinality $ω_1$, is $z$-embedded in the product $βX\times βD(ω_1)$ of the Stone--Cech compactifications. It is also proved that $\mathbb R^{ω_1}$-factorizability is hereditary and countably multiplicative, that any $\mathbb R^{ω_1}$-factorizable space is hereditarily Lindelöf and hereditarily separable, and that the existence of nonmetrizable $\mathbb R^{ω_1}$-factorizable topological spaces and groups is independent of ZFC: under CH, all $\mathbb R^{ω_1}$-factorizable spaces are second-countable, while under MA + $\lnot$CH, the countable Fréchet--Urysohn fan is $\mathbb R^{ω_1}$-factorizable.