  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 $\widetilde W$-spaces, and we provide several
applications. Namely, we use the notion of $\widetilde W$-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.
