The $d^{*}$-space

TL;DR

This paper introduces the concept of $d^{*}$-spaces in topology, explores their properties, characterizations, and relationships with function and power spaces, expanding the understanding of these spaces in topological theory.

Contribution

The paper defines $d^{*}$-spaces, characterizes them, and investigates their stability under retracts and their behavior in function and power space constructions.

Findings

Strong $d$-spaces are $d^{*}$-spaces, but not vice versa.

If the Isbell topology on $TOP(X,Y)$ is a $d^{*}$-space, then $Y$ is a $d^{*}$-space.

If the Smyth power space $Q_{v}(X)$ is a $d^{*}$-space, then $X$ is a $d^{*}$-space.

Abstract

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.