The $G$-connected property and $G$-topological groups

TL;DR

This paper explores properties of $G$-connectedness and $G$-topological groups, demonstrating the preservation of $G$-connectedness under countable products and defining $G$-topological groups with specific regularity conditions.

Contribution

It introduces the concept of $G$-topological groups and extends existing results on $G$-connectedness, including preservation under countable products.

Findings

$G$-connectedness is preserved by countable products

Introduction of $G$-topological groups

Establishment of conditions for $G$-topology in groups

Abstract

In this paper, we discuss some properties of of $G$-hull, $G$-kernel and $G$-connectedness, and extend some results of \cite{life34}. In particular, we prove that the $G$-connectedness are preserved by countable product. Moreover, we introduce the concept of $G$-topological group, and prove that a $G$-topological group is a $G$-topology under the assumption of the regular method preserving the subsequence.