$\mathcal I^K$-limit points, $\mathcal I^K$-cluster points and $\mathcal I^K$-Frechet compactness

Abstract

In 2011, the theory of $\mathcal I^K$-convergence gets birth as an extension of the concept of $\mathcal{I}^*$-convergence of sequences of real numbers. $\mathcal I^K$-limit points and $\mathcal I^K$-cluster points of functions are introduced and studied to some extent, where $\mathcal{I}$ and $\mathcal{K}$ are ideals on a non-empty set $S$. In a first countable space set of $\mathcal I^K$-cluster points is coincide with the closure of all sets in the filter base $\mathcal{B}_f(\mathcal{I^K})$ for some function $f : S\to X$. Frechet compactness is studied in light of ideals $\mathcal{I}$ and $\mathcal{K}$ of subsets of $S$ and showed that in $\mathcal{I}$-sequential $T_2$ space Frechet compactness and $\mathcal{I}$-Frechet compactness are equivalent. A class of ideals have been identified for which $\mathcal I^K$-Frechet compactness coincides with $\mathcal{I}$-Frechet compactness in first countable spaces.