${\mathcal{I}}^K$-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 ${\mathcal{I}}^{\ast }$-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 ${\mathcal{B}}_f({\mathcal{I}}^{\mathcal{K}})$ 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.

14 pages, 1 figures