Abstract

Notions of $\mathcal{I}^\mathcal{K}$-limit points and $\mathcal{I}^\mathcal{K}$-cluster points of functions are studied in topological spaces. In the first countable space, all $\mathcal{I}^\mathcal{K}$-cluster points of a function $f:S\to X$ belong to the closure of each member of the filter base $\mathcal{B}_f(\mathcal{I}^\mathcal{K})$. Fréchet compactness is studied in the light of ideals $\mathcal{I}$ and $\mathcal{K}$ of subsets of $S$ and showed that in an $\mathcal{I}$-sequential Hausdorff space, Fréchet compactness and $\mathcal{I}$-Fréchet compactness are equivalent. Using the FDS-property introduced by D. Shakmatov, M. Tkachenko, R. Wilson in Houston J. Math., it is seen that $\mathcal{I}^\mathcal{K}$-Fréchet compactness, Fréchet compactness and $\mathcal{I}$-Fréchet compactness are equivalent for a particular class of ideals on $S$.

Keywords: Ideal; $\mathcal{I}$-nonthin; $\mathcal{I}$-Fréchet compactness; $\mathcal{I}^\mathcal{K}$-Fréchet compactness.

MSC: 54D30, 54A05

DOI: 10.57016/MV-MTZE3327

Pages:  1--12