Further studies on open well-filtered spaces

TL;DR

This work investigates open well-filtered spaces, focusing on whether the property is preserved by the upper space construction $\mathcal{D}(X)$ (the Smyth upper space with the upper Vietoris topology) and by related operations. It introduces OWF-sets and establishes a characterization: a $T_0$ space $X$ is open well-filtered iff every $A$ in $\mathsf{OWF}(X)$ is of the form $A=cl_X(\{x\})$ for a unique $x\in X$; this parallels KF-set characterizations for well-filtered spaces. The main result proves that open well-filteredness is preserved by passing to the upper space, and the paper also discusses limitations via examples (saturated subspaces, retracts, and certain products) and provides an open problem: whether the converse implication holds, i.e., whether $\mathcal{D}(X)$ being open well-filtered implies that $X$ is open well-filtered. Overall, the results advance the understanding of how open well-filteredness interacts with standard constructions and its relation to soberness and core-compactness.

Abstract

The open well-filtered spaces were introduced by Shen, Xi, Xu and Zhao to answer the problem whether every core-compact well-filtered space is sober. In the current paper we explore further properties of open well-filtered spaces. One of the main results is that if a space is open well-filtered, then so is its upper space (the set of all nonempty saturated compact subsets equipped with the upper Vietoris topology). Some other properties on open well-filtered spaces are also studied.

Figures (2)

• Figure 1: The Johnstone's dcpo $\mathbb J$
• Figure 2: The poset $P$ of Example \ref{['exam']}

Theorems & Definitions (32)

• remark 1
• lemma 1
• definition 1
• proposition 1
• definition 2
• remark 2
• theorem 1
• definition 3
• definition 4
• proposition 2