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

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.