The $d^{*}$-space

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.