$P$-spaces in the absence of the Axiom of Choice

TL;DR

This paper explores the properties of $P$-spaces within set theory frameworks lacking the Axiom of Choice, introducing new weaker axioms, proving independence results, and posing open problems about their behavior.

Contribution

It introduces new weaker forms of the Axiom of Choice relevant to $P$-spaces and countable intersections of $G_{δ}$-sets, and investigates their properties without the Axiom of Choice.

Findings

A zero-dimensional subspace of the real line may not be strongly zero-dimensional in ZF.

Several independence results regarding $P$-spaces are established.

Open problems about finite products of $P$-spaces are posed and partially addressed.

Abstract

A $P$-space is a topological space whose every $G_{\delta}$-set is open. In this article, basic properties of $P$-spaces are investigated in the absence of the Axiom of Choice. New weaker forms of the Axiom of Choice, all relevant to $P$-spaces or to countable intersections of $G_{\delta}$-sets, are introduced. Several independence results are obtained and open problems are posed. It is shown that a zero-dimensional subspace of the real line may fail to be strongly zero-dimensional in $\mathbf{ZF}$. Among the open problems there is the question whether it is provable in $\mathbf{ZF}$ that every finite product of $P$-spaces is a $P$-space. A partial answer to this question is given.