Further aspects of $\mathcal{I}^{\mathcal{K}}$-convergence in Topological Spaces

TL;DR

This paper explores advanced ideal convergence modes in topological spaces, introduces a new class called alI^K-sequential spaces, and characterizes alI^K-cluster points as closed sets.

Contribution

It introduces the alI^K-sequential space and extends the theory of ideal convergence modes in topological spaces.

Findings

Relationships between various ideal convergence modes established.

alI^K-sequential spaces contain all sequential spaces.

Set of alI^K-cluster points characterized as closed sets.

Abstract

In this paper, we obtain some results on the relationships between different ideal \linebreak convergence modes namely, $\mathcal{I}^\mathcal{K}$, $\mathcal{I}^{\mathcal{K}^*}$, $\mathcal{I}$, $\mathcal{K}$, $\mathcal{I} \cup \mathcal{K}$ and $(\mathcal{I} \cup \mathcal{K})^*$. We introduce a topological space namely $\mathcal{I}^\mathcal{K}$-sequential space and show that the class of $\mathcal{I}^\mathcal{K}$-sequential spaces contain the sequential spaces. Further $\mathcal{I}^\mathcal{K}$-notions of cluster points and limit points of a function are also introduced here. For a given sequence in a topological space $X$, we characterize the set of $\mathcal{I}^\mathcal{K}$-cluster points of the sequence as closed subsets of $X$.