The $d^{\ast }$-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.