Constructing crowded Hausdorff $P$-spaces in set theory without the axiom of choice

Abstract

For an infinite set $X$, a closed under finite unions family $\mathcal{Z}$ with $[X{]}^{<\omega }\subseteq \mathcal{Z}\subseteq \mathcal{P}(X)$, and any $\mathcal{A}\subseteq \mathcal{P}(X)$, the topology ${\tau }_{\mathcal{A}}[\mathcal{Z}]=\{V\in \mathcal{A}:(\forall x\in V)(\exists z\in \mathcal{Z})(x\cap z=\varnothing \wedge \{y\in \mathcal{A}:x\subseteq y\subseteq X\setminus z\}\subseteq V)\}$ 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 $\mathbf{S}(X,[X{]}^{\le \omega })=⟨\mathcal{A},{\tau }_{\mathcal{A}}[\mathcal{Z}]⟩$ for $\mathcal{A}=[X{]}^{<\omega }$ and $\mathcal{Z}=[X{]}^{\le \omega }$ are of special importance here. Among many other results, the following theorems are proved in $\mathbf{ZF}$: (1) If $X$ is uncountable, then $\mathbf{S}(X,[X{]}^{\le \omega })$ is a crowded zero-dimensional Hausdorff space, and if $X$ is also quasi Dedekind-finite, then $\mathbf{S}(X,[X{]}^{\le \omega })$ is a $P$-space; (2) $\mathbf{S}({\omega }_1,[{\omega }_1{]}^{\le \omega })$ is a $P$-space if and only if ${\omega }_1$ is regular; (3) the axiom of countable choice for families of finite sets is equivalent to the statement ``for every infinite Dedekind-finite set $X$, $\mathbf{S}(X,[X{]}^{\le \omega })$ is a $P$-space''; (4) the statement ``$\mathbb{R}$ admits a topology $\tau$ such that $⟨\mathbb{R},\tau ⟩$ 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 ``${\omega }_1$ is regular''. A lot of relevant independence results are obtained.