JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2026, Vol. 61 ›› Issue (4): 62-68.doi: 10.6040/j.issn.1671-9352.0.2024.279

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Notes on the continuity of directed spaces

WANG Wu   

  1. Zhonghuan Information College Tianjin University of Technology, Tianjin 300380, China
  • Published:2026-04-08

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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 powerspace. 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 powerspace of a directed space is continuous, then it is meet continuous. Furthermore, a directed space is continuous if and only if its directed probability powerspace is continuous.

Key words: directed space, meet continuous space, one step closure, continuous space, directed probability powerspace

CLC Number: 

  • O153
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