G-Connectedness in Topological Groups with Operations

Abstract

It is a well known fact that for a Hausdorff topological group $X$, the limits  of convergent sequences in $X$ define a function denoted by $\lim$  from the set of all convergent sequences in $X$ to $X$.  This notion has been modified by Connor and Grosse-Erdmann for real functions by replacing $\lim$ with an arbitrary linear functional $G$ defined on a linear subspace of the vector space of all real sequences. Recently  some authors have  extended the concept to the topological group setting and introduced  the concepts  of $G$-sequential continuity,  $G$-sequential compactness and $G$-sequential connectedness.  In this paper we  present some results about  $G$-sequentially closures,  $G$-sequentially connectedness  and fundamental system of $G$-sequentially open neighbourhoods for  a wide class of topological algebraic structures called groups with operations, which include topological groups, topological rings without identity, {R}-modules,  Lie algebras, Jordan algebras, and many others.