On a certain generalization of $W$-spaces

Abstract

We present a simple generalization of $W$-spaces introduced by G. Gruenhage. We show that this generalization leads to a strictly larger class of topological spaces which we call $\tilde{W}$-spaces, and we provide several applications. Namely, we use the notion of $\tilde{W}$-spaces to provide sufficient conditions for the product of two spaces to be a Baire space, for a semitopological group to be a topological group, or for a separately continuous function to be continuous at the points of a certain large set.