关于定向空间连续性的注记

Notes on the continuity of directed spaces

摘要

定向空间的交连续性与连续性是研究拓扑空间的重要性质,本文给出一步闭包和弱一步闭包的概念,并研究它们与交连续空间的关系。利用定向扩展概率幂空间给出连续空间的等价刻画。如果定向空间有一步闭包,则其是交连续空间,定向空间是有弱一步闭包的交连续空间当且仅当其有一步闭包。如果定向空间的定向扩展概率幂空间连续,则定向空间是交连续的,定向空间是连续空间当且仅当其定向扩展概率幂空间是连续空间。

Abstract

The continuity and meet continuity of directed spaces are important properties to study topological space. We give the concepts of one-step closure and weak one-step closure, and study their relationship with meet continuous spaces. An equivalent characterization of a continuous space is given using the directed probability power space. If the directed space has a one-step closure, then it is meet continuous space. The directed space is an meet continuous space with a weak one-step closure if and only if it has a one-step closure. If the directed probability power space of a directed space is continuous, then it is meet continuous. Furthermore, a directed space is continuous if and only if its directed probability power space is continuous.

关键词

定向空间 / 交连续空间 / 一步闭包 / 连续空间 / 定向扩展概率幂空间

Key words

directed space / meet continuous space / one step closure / continuous space / directed probability power space

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基金资助

天津市教委科研计划项目(2023KJ281)