On $P$-spaces and $G_{\delta}$-sets in the absence of the Axiom of Choice

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 for applications. Special subrings of rings of continuous real functions are applied. New notions of a quasi Baire space and a strongly (quasi) Baire space are introduced. Several independence results are obtained. For instance, it is shown in $\mathbf{ZF}$ that if $G_{\delta}$-modifications of Tychonoff spaces are $P$-spaces, then every denumerable family of denumerable sets has a multiple choice function. In $\mathbf{ZF}$, a zero-dimensional subspace of $\mathbb{R}$ may fail to be strongly zero-dimensional, and countable intersections of $G_{\delta}$-sets of $\mathbb{R}$ may fail to be $G_{\delta}$-sets. New open problems are posed. Partial answers to some of them are given.