Classes of Baire spaces defined by semi-neighborhoods of the diagonal

TL;DR

This paper introduces new classes of Baire spaces using semi-neighborhoods of the diagonal and explores their properties and implications for continuity in various types of topological groups.

Contribution

It defines the classes $ ext{Delta}$, $ ext{Delta}_h$, and $ ext{Delta}_s$ Baire spaces using semi-neighborhoods and studies their properties through topological games, linking them to group continuity.

Findings

Classes of Baire spaces are characterized by semi-neighborhoods of the diagonal.

These classes help in understanding the continuity properties of different topological groups.

Topological groups with certain Baire space properties are shown to be topological groups.

Abstract

With the help of semi-neighborhoods of the diagonal, classes of Baire spaces are defined: $\Delta$, $\Delta_h$ and $\Delta_s$ Baire spaces. These classes of spaces are studied with the help of topological games. They are useful in studying continuity in groups: paratopological $\Delta$-Baire groups, quasi-topological $\Delta_h$-Baire groups, and semitopological $\Delta_s$-Baire groups are topological groups.