Ankur Sharmah | Searcharxiv

arXiv2021

On $I^K$-Convergence in a Topological space via semi-open sets

In this article, we consider $\mathcal{I}^\mathcal{K}$-convergence to define a new concept of convergence namely, $\mathcal{S}$-$\mathcal{I}^\mathcal{K}$-convergence which generalizes the notion of $\mathcal{S}$-$\mathcal{I}$-convergence introduced by Guevara et al. \cite{GSR20} recently. Some properties of $\mathcal{S}$-$\mathcal{I}^\mathcal{K}$-convergence of sequences and its relation with compact sets are discussed. In particular, we investigate the relation between semi-compactness and semi-Lindeloffness by introducing the notion of $\mathcal{S}$-$\mathcal{I}^\mathcal{K}$-cluster point of a sequence. The "Equivalence between semi-dense and dense sets" is utilized to characterize the set of $\mathcal{S}$-$\mathcal{I}^\mathcal{K}$-cluster points of a sequence as semi-closed subsets of a topological space. Moreover, in product space, we obtain some results for $\mathcal{I}^\mathcal{K}$-convergence which also holds for $\mathcal{S}$-$\mathcal{I}^\mathcal{K}$-convergence.

math.GN↗

arXiv2021

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

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 $X$.

math.GN↗