On covering and quotient maps for $\mathcal{I}^{\mathcal{K}}$-convergence in topological spaces
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In this article, we show that the family of all $\mathcal{I}^\mathcal{K}$-open subsets in a topological space forms a topology if $\mathcal{K}$ is a maximal ideal. We introduce the notion of $\mathcal{I}^\mathcal{K}$-covering map and investigate some basic properties. The notion of quotient map is studied in the context of $\mathcal{I}^\mathcal{K}$-convergence and the relationship between $\mathcal{I}^\mathcal{K}$-continuity and $\mathcal{I}^\mathcal{K}$-quotient map is established. We show that for a maximal ideal $\mathcal{K}$, the properties of continuity and preserving $\mathcal{I}^\mathcal{K}$-convergence of a function defined on $X$ coincide if and only if $X$ is an $\mathcal{I}^\mathcal{K}$-sequential space.
Abstract
In this article, we show that the family of all $\mathcal{I}^\mathcal{K}$-open subsets in a topological space forms a topology if $\mathcal{K}$ is a maximal ideal. We introduce the notion of $\mathcal{I}^\mathcal{K}$-covering map and investigate some basic properties. The notion of quotient map is studied in the context of $\mathcal{I}^\mathcal{K}$-convergence and the relationship between $\mathcal{I}^\mathcal{K}$-continuity and $\mathcal{I}^\mathcal{K}$-quotient map is established. We show that for a maximal ideal $\mathcal{K}$, the properties of continuity and preserving $\mathcal{I}^\mathcal{K}$-convergence of a function defined on $X$ coincide if and only if $X$ is an $\mathcal{I}^\mathcal{K}$-sequential space.
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