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 $(X,\ddot{I}„)$ is a strong $R$-space iff every nonempty $\ddot{I}„$-closed subset of $X$ is compact in $(X,\ddot{I}{„}^d)$, where $\ddot{I}{„}^d$ is the de Groot dual of $\ddot{I}„$; consequently, if $(X,\ddot{I}„)$ is a strong $R$-space (especially, if $(X,\ddot{I}„)$ is a coherent well-filtered space), then $\ddot{I}„\subseteq \ddot{I}{„}^{dd}$. Therefore, for any locally compact strong $R$-space $(X,\ddot{I}„)$, we have $\ddot{I}„=\ddot{I}{„}^{dd}$. 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.

16 pages, 5 figures