On G-Sequential Continuity

Let $X$ be a first countable Hausdorff topological group. The limit of a sequence in $X$ defines a function denoted by $lim$ from the set of all convergence sequences to $X$. This definition was 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. \c{C}akall{\i} extended the concept to topological group setting and introduced the concept of $G$-sequential compactness and investigated $G$-sequential continuity and $G$-sequential compactness in topological groups. In this paper we give a further investigation of $G$-sequential continuity in topological groups most of which are also new for the real case.

💡 Research Summary

The paper investigates the notion of G‑sequential continuity in the setting of first‑countable Hausdorff topological groups. Starting from the classical limit functional “lim” on convergent sequences, the authors adopt the framework introduced by Connor and Grosse‑Erdmann, replacing lim by an arbitrary linear functional G defined on a linear subspace c_G(X) of the space s(X) of all X‑valued sequences. The group X is assumed to be a Hausdorff topological group with a countable local base, written additively.

Basic definitions.
A sequence x ∈ c_G(X) is called G‑convergent to ℓ if G(x)=ℓ. The method G is called regular when every ordinary convergent sequence is also G‑convergent and G coincides with the ordinary limit on such sequences. The G‑sequential closure of a set A⊂X is \