Sequential Definitions of Connectedness

TL;DR

This paper explores how different definitions of sequence convergence affect the concept of connectedness in topological groups, generalizing classical notions through $G$-sequential closure.

Contribution

It introduces a generalized framework for sequential connectedness in topological groups based on various convergence definitions, extending classical topological concepts.

Findings

Impact of different $G$-convergence on connectedness structure

Generalization of sequential connectedness via $G$-sequential closure

Special case of classical connectedness when G = lim

Abstract

A topological group $X$ is called connected if the only subsets which are both open and closed are the whole space $X$ and the null set $\emptyset$. A subset of a topological group is connected if the subspace is connected. We say that a subset $A$ of $X$ is $G$-sequentially connected if the only subsets of $A$ which are both $G$-sequentially open and $G$-sequentially closed, with respect to the relative $G$-sequentially open and $G$-sequentially closed subsets of $A$, are open and closed subsets of $A$ are $A$ and the null set, $\emptyset$. We investigate the impact of changing the definition of convergence of sequences on the structure of sequential connectedness of subsets of $X$ via sequential closure of sets in the sense of $G$-sequential closure. Sequential connectedness for topological groups is a special case of this generalization when G = lim.