Constructing crowded Hausdorff $P$-spaces in set theory without the axiom of choice
Eleftherios Tachtsis, Eliza Wajch
TL;DR
This work addresses the existence and construction of nonempty crowded zero-dimensional Hausdorff P-spaces within ZF (without AC) by introducing a convex-topology framework on families of subsets and analyzing spaces 𝕊_A(X,𝒵). The authors develop two parallel constructions, 𝕊_A(X,𝒵) and 2^X[𝒵], linking their P-space-ness to weak-choice principles such as CAC and CAC_fin, and establish exact equivalences in several key cases (e.g., CAC ⇔ all 𝕊_𝒫(X,[X]^{≤ω}) are P-spaces; ω_1 regularity ↔ 𝕊_𝕡(ω_1,[ω_1]^{≤ω}) is P-space). They derive numerous independence results using a range of permutation and symmetric models (e.g., N2, N18, N53, M1, M9), demonstrating that many choiceless topological phenomena can occur independently of AC and depending on the chosen weak-choice framework. The paper also introduces new PS_0–PS_5 principles, proves their connections to CAC_fin, and analyzes their relative strength, yielding a rich landscape of interdependent combinatorial-topological statements. Overall, the results significantly advance the understanding of constructing crowded zero-dimensional P-spaces without the Axiom of Choice and map the boundaries between topology and weak set-theoretic choice principles.
Abstract
For an infinite set $X$, a closed under finite unions family $\mathcal{Z}$ with $[X]^{<ω}\subseteq\mathcal{Z}\subseteq\mathcal{P}(X)$, and any $\mathcal{A}\subseteq\mathcal{P}(X)$, the topology $τ_{\mathcal{A}}[\mathcal{Z}]=\{V\in\mathcal{A}: (\forall x\in V)(\exists z\in \mathcal{Z})(x\cap z=\emptyset \wedge \{y\in\mathcal{A}: x\subseteq y\subseteq X\setminus z\}\subseteq V)\}$ on $\mathcal{A}$ is investigated to give answers to the following open problem in various models of $\mathbf{ZF}$ or $\mathbf{ZFA}$: Is there a non-empty Hausdorff, crowded zero-dimensional $P$-space in the absence of the axiom of choice? Spaces of the form $\mathbf{S}(X, [X]^{\leqω})=\langle \mathcal{A}, τ_{\mathcal{A}}[\mathcal{Z}]\rangle$ for $\mathcal{A}=[X]^{<ω}$ and $\mathcal{Z}=[X]^{\leqω}$ are of special importance here. Among many other results, the following theorems are proved in $\mathbf{ZF}$: (1) If $X$ is uncountable, then $\mathbf{S}(X, [X]^{\leqω})$ is a crowded zero-dimensional Hausdorff space, and if $X$ is also quasi Dedekind-finite, then $\mathbf{S}(X, [X]^{\leqω})$ is a $P$-space; (2) $\mathbf{S}(ω_1, [ω_1]^{\leqω})$ is a $P$-space if and only if $ω_1$ is regular; (3) the axiom of countable choice for families of finite sets is equivalent to the statement ``for every infinite Dedekind-finite set $X$, $\mathbf{S}(X,[X]^{\leqω})$ is a $P$-space''; (4) the statement ``$\mathbb{R}$ admits a topology $τ$ such that $\langle\mathbb{R}, τ\rangle$ is a crowded, zero-dimensional Hausdorff $P$-space'' is strictly weaker than the axiom of countable choice for families of subsets of $\mathbb{R}$; (5) the statement ``there exists a non-empty, well-orderable crowded zero-dimensional Hausdorff $P$-space'' is strictly weaker than ``$ω_1$ is regular''. A lot of relevant independence results are obtained.