AI 中文总结
研究强R空间基本性质，包括其遗传性、可收缩性等，证明T0空间是强R空间的充要条件，探讨斯迈斯幂空间和斯科特幂空间成为强R空间的条件，揭示强R空间在相关范畴中的特性及不同空间间关系。
AI 中文摘要
本文主要研究强R空间的一些基本性质。证明了强R空间的性质是闭遗传、饱和遗传和可收缩的，但不是有限可积的。得出强R空间和连续映射的范畴在T0空间和连续映射的范畴中不是反射的。证明了T0空间(X,τ)是强R空间当且仅当X的每个非空τ闭子集在(X,τd)中是紧的，其中τd是τ的德格罗对偶。最后研究了T0空间的斯迈斯幂空间和斯科特幂空间是强R空间的条件并给出了几个这样的条件。
英文摘要
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 $(X, τ)$ is a strong $R$-space iff every nonempty $τ$-closed subset of $X$ is compact in $(X, τ^{d})$, where $τ^d$ is the de Groot dual of $τ$; consequently, if $(X, τ)$ is a strong $R$-space (especially, if $(X, τ)$ is a coherent well-filtered space), then $τ\subseteq τ^{dd}$. Therefore, for any locally compact strong $R$-space $(X, τ)$, we have $τ=τ^{dd}$. 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.
Comments16 pages, 5 figures
