On P-spaces and Gδ-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 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 ZF, a zero-dimensional subspace of R may fail to be strongly zero-dimensional, and countable intersections of G(delta)-sets of R may fail to be G(delta)-sets. New open problems are posed. Partial answers to some of them are given.