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Ural Math. J., 2023 Volume 9, Issue 2, Pages 46–59 (Mi umj203)

$\mathcal{I}^{\mathcal{K}}$-sequential topology

H. S. Behmanush, M. Küçükaslan

Mersin Üniversitesi

Abstract: In the literature, $\mathcal{I}$-convergence (or convergence in $\mathcal{I}$) was first introduced in [11].
Later related notions of $\mathcal{I}$-sequential topological space and $\mathcal{I}^*$-sequential topological space were introduced and studied. From the definitions it is clear that $\mathcal{I}^*$-sequential topological space is larger(finer) than $\mathcal{I}$-sequential topological space. This rises a question: is there any topology (different from discrete topology) on the topological space $\mathcal{X}$ which is finer than $\mathcal{I}^*$-topological space? In this paper, we tried to find the answer to the question. We define $\mathcal{I}^{\mathcal{K}}$-sequential topology for any ideals $\mathcal{I}$, $\mathcal{K}$ and study main properties of it. First of all, some fundamental results about $\mathcal{I}^{\mathcal{K}}$-convergence of a sequence in a topological space $(\mathcal{X} ,\mathcal{T})$ are derived. After that, $\mathcal{I}^{\mathcal{K}}$-continuity and the subspace of the $\mathcal{I}^{\mathcal{K}}$-sequential topological space are investigated.

Keywords: ideal convergence, $\mathcal{I}^{\mathcal{K}}$-convergence, sequential topology, $\mathcal{I}^{\mathcal{K}}$-sequential topology.

Language: English

DOI: 10.15826/umj.2023.2.004



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© Steklov Math. Inst. of RAS, 2024