Abstract

A sequence $\{x_n\}$ is $\mathcal{S}$-$\mathcal{I}^\mathcal{K}$-convergent to $\xi$, if there exists a "big enough" subsequence $\{x_{n_k}\}$ which $\mathcal{K}$-converges to $\xi$ via semi-open sets. In this paper, we introduce the concept of $\mathcal{S}$-$\mathcal{I}^\mathcal{K}$-convergence which generalizes $\mathcal{S}$-$\mathcal{I}$-convergence and discuss some properties, as well as its relation with compact sets. For two given ideals $\mathcal{I}$ and $\mathcal{K}$, we justify the existence of an ideal such that $\mathcal{I}^\mathcal{K}$-convergence and convergence with the third ideal coincides for semi-open sets. Moreover, the notion of $\mathcal{S}$-$\mathcal{I}^\mathcal{K}$-cluster point of a sequence is defined and studied here. We characterize the collection of $\mathcal{S}$-$\mathcal{I}^\mathcal{K}$-cluster points of a sequence as semi-closed subsets of the space.

Keywords: $\mathcal{I}^\mathcal{K}$-convergence; $\mathcal{S}$-$\mathcal{I}^\mathcal{K}$-convergence; $\mathcal{S}$-$\mathcal{I}^{\mathcal{K}}$-cluster points; semi-open sets; semi-compactness; semi-dense set.

MSC: 40A35, 40A05, 54A05, 54A20

DOI: 10.57016/MV-tA497YI4

Pages:  1--13