Almost-P-Spaces

A P-space is a topological space in which every Gδ-set is open. P -spaces are fairly rare. For example, the only compact (or even pseudocompact) P-spaces are finite. A larger class of spaces, the almost-P-spaces, consists of those spaces in which G δ-sets have dense interiors. The almost-P-spaces are far less restricted than the P -spaces—for example, there are infinite, compact, connected almost- P -spaces. In this paper, we study almost-P-spaces and raise a number of questions relating to them.