Xiangping Chu ; Qingguo Li - The $d^{*}$-space

entics:10354 - Electronic Notes in Theoretical Informatics and Computer Science, March 21, 2023, Volume 2 - Proceedings of ISDT 9 - https://doi.org/10.46298/entics.10354
The $d^{*}$-spaceArticle

Authors: Xiangping Chu Qingguo Li

    In this paper, we introduce the concept of $d^{\ast}$-spaces. We find that strong $d$-spaces are $d^{\ast}$-spaces, but the converse does not hold. We give a characterization for a topological space to be a $d^{\ast}$-space. We prove that the retract of a $d^{\ast}$-space is a $d^{\ast}$-space. We obtain the result that for any $T_{0}$ space $X$ and $Y$, if the function space $TOP(X,Y)$ endowed with the Isbell topology is a $d^{\ast}$-space, then $Y$ is a $d^{\ast}$-space. We also show that for any $T_{0}$ space $X$, if the Smyth power space $Q_{v}(X)$ is a $d^{\ast}$-space, then $X$ is a $d^{\ast}$-space. Meanwhile, we give a counterexample to illustrate that conversely, for a $d^{\ast}$-space $X$, the Smyth power space $Q_{v}(X)$ may not be a $d^{\ast}$-space.

    Volume: Volume 2 - Proceedings of ISDT 9
    Published on: March 21, 2023
    Accepted on: November 25, 2022
    Submitted on: November 23, 2022
    Keywords: Mathematics - General Topology

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