On $I^K$-Convergence in a Topological space via semi-open sets

TL;DR

This paper introduces a new form of convergence called $\\mathcal{S}$-$\\\mathcal{I}^\\\mathcal{K}$-convergence in topological spaces, exploring its properties, relation to compactness, and behavior in product spaces, extending previous convergence concepts.

Contribution

It defines and studies $\\mathcal{S}$-$\\mathcal{I}^\\mathcal{K}$-convergence, generalizing existing notions and analyzing its properties and implications in topology.

Findings

Characterizes $\\mathcal{S}$-$\\mathcal{I}^\\mathcal{K}$-cluster points as semi-closed sets.

Establishes relations between semi-compactness and semi-Lindeloffness.

Extends convergence results to product spaces.

Abstract

In this article, we consider $\mathcal{I}^\mathcal{K}$-convergence to define a new concept of convergence namely, $\mathcal{S}$-$\mathcal{I}^\mathcal{K}$-convergence which generalizes the notion of $\mathcal{S}$-$\mathcal{I}$-convergence introduced by Guevara et al. \cite{GSR20} recently. Some properties of $\mathcal{S}$-$\mathcal{I}^\mathcal{K}$-convergence of sequences and its relation with compact sets are discussed. In particular, we investigate the relation between semi-compactness and semi-Lindeloffness by introducing the notion of $\mathcal{S}$-$\mathcal{I}^\mathcal{K}$-cluster point of a sequence. The "Equivalence between semi-dense and dense sets" is utilized to characterize the set of $\mathcal{S}$-$\mathcal{I}^\mathcal{K}$-cluster points of a sequence as semi-closed subsets of a topological space. Moreover, in product space, we obtain some results for $\mathcal{I}^\mathcal{K}$-convergence which also holds for $\mathcal{S}$-$\mathcal{I}^\mathcal{K}$-convergence.