On a certain generalization of W-spaces

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 W˜-spaces, and we provide several applications. Namely, we use the notion of 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.