S*-well-filtered spaces and d*-spaces

TL;DR

This paper introduces S*-well-filtered spaces, a new class of T0-spaces that extends strongly well-filtered spaces, explores their properties, and distinguishes them from related classes like d*-spaces and weak well-filtered spaces.

Contribution

It defines S*-well-filtered spaces, establishes their relationship with d*-spaces, and demonstrates key properties and distinctions from strongly well-filtered spaces.

Findings

S*-well-filtered spaces are strictly larger than strongly well-filtered spaces.

Scott space P is d*-space iff it is S*-well-filtered.

Johnstone's non-sober dcpo is S*-well-filtered but not strongly well-filtered.

Abstract

Recently, Xu proposed a strongly well-filtered space in [24] and systematically investigated some of its properties and characterizations. In this paper, we introduce a new class of T0-spaces called S*-well-filtered spaces, which is strictly larger than the class of strongly well-filtered spaces. First, we establish some connections among S*-well-filtered spaces, d*-spaces and weak well-filtered spaces. Then it is demonstrated that for any dcpo P, the Scott space {\Sigma}P is a d*-space if and only if it is S*-well-filtered. Furthermore, some basic properties of S*-well-filtered spaces are discussed. We prove that if Y is an S*-well-filtered space, the function space TOP(X,Y) equipped with the Isbell topology may not be an S*-well-filtered space. Finally, we study the S*-well-filteredness of Smyth power spaces. In addition, Johnstone's non-sober dcpo example is shown to be S*-well-filtered yet it is not strongly well-filtered, thereby establishing an obvious distinction between these two classes of dcpos.