\mathbb R^ω₁-Factorizable Spaces and Groups

2025/09/05 by A. E. Lipin, Anton Lipin, Evgenii Reznichenko +5 · 1 voice
Mathematics · #22A05 #54F45 #Advanced Banach Space Theory #Advanced Harmonic Analysis Research #Advanced Topology and Set Theory #FOS: Mathematics #General Topology (math.GN) #math.GN

paper · pdf · doi:10.48550/arxiv.2509.05105

openalex publication_date 2025/09/05 · arxiv published 2025/09/05 · openalex created_date 2025/10/10 · arxiv updated 2026/01/20 · openalex updated_date 2026/07/28

Abstract

A topological space X is \mathbb R^ω₁-factorizable if any continuous function f\colon X→ \mathbb R^ω₁ factors through a continuous function from X to a second-countable space. It is shown that a Tychonoff space X is \mathbb R^ω₁-factorizable if and only if X× D(ω₁), where D(ω₁) is a discrete space of cardinality ω₁, is z-embedded in the product βX× βD(ω₁) of the Stone--Cech compactifications. It is also proved that \mathbb R^ω₁-factorizability is hereditary and countably multiplicative, that any \mathbb R^ω₁-factorizable space is hereditarily Lindelöf and hereditarily separable, and that the existence of nonmetrizable \mathbb R^ω₁-factorizable topological spaces and groups is independent of ZFC: under CH, all \mathbb R^ω₁-factorizable spaces are second-countable, while under MA + ¬CH, the countable Fréchet--Urysohn fan is \mathbb R^ω₁-factorizable.

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

• Anton Lipin, Evgenii Reznichenko, Ol'ga Sipacheva: $\mathbb R^{\omega_1}$-Factorizable Spaces and Groups https://arxiv.org/abs/2509.05105 https://arxiv.org/pdf/2509.05105 https://arxiv.org/html/2509.0 [bsky, 0 points, 0 comments]

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