# On a certain generalization of $W$-spaces

## TL;DR

This paper introduces a broader class of topological spaces called W-spaces, called W-spaces, and demonstrates their usefulness in various topological and group-theoretic contexts.

## Contribution

It generalizes the concept of W-spaces to W-spaces, expanding the class of spaces with useful properties and applications.

## Key findings

- Provides conditions for product spaces to be Baire spaces
- Establishes when semitopological groups are topological groups
- Identifies criteria for separate continuity implying continuity

## 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 $\widetilde W$-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.