On strong $R$-spaces

Abstract

In this paper, we mainly investigate some basic properties of strong $R$-spaces. It is shown that the property of being a strong $R$-space is closed-hereditary, saturated-hereditary and retractive, but not finite productive. Hence the category $\mathbf{S}$-$\mathbf{Top}_r$ of strong $R$-spaces and continuous mappings is not reflective in the category $\mathbf{Top}_0$ of $T_0$-spaces and continuous mappings. It is proved that a $T_0$-space $(X, \tau)$ is a strong $R$-space iff every nonempty $\tau$-closed subset of $X$ is compact in $(X, \tau^{d})$, where $\tau^d$ is the de Groot dual of $\tau$; consequently, if $(X, \tau)$ is a strong $R$-space (especially, if $(X, \tau)$ is a coherent well-filtered space), then $\tau \subseteq \tau^{dd}$. Therefore, for any locally compact strong $R$-space $(X, \tau)$, we have $\tau=\tau^{dd}$. Finally, we investigate conditions under which the Smyth power space and Scott power space of a $T_0$-space is a strong $R$-space. Several such conditions are given.