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

Nana Han, Siheng Chen, Qingguo Li

TL;DR

The paper defines $S^{\ast}$-well-filtered spaces as a strictly weaker analogue of strongly well-filtered spaces and analyzes their position relative to $d^{\ast}$-spaces and weak well-filtered spaces. It proves that for any dcpo $P$, the Scott space $\Sigma P$ is a $d^{\ast}$-space if and only if it is $S^{\ast}$-well-filtered, and it exhibits Johnstone's non-sober dcpo as $S^{\ast}$-well-filtered yet not strongly well-filtered, illustrating a clear separation of classes. The authors develop a broad suite of properties, including closure under subspaces, behavior under retracts, and nuanced product behavior; they also study the effect of Isbell-topologized function spaces and Smyth power spaces on $S^{\ast}$-well-filteredness, establishing several transfer principles and counterexamples. Together, these results refine the hierarchy of domain-theoretic space classes and clarify how power-space constructions interact with $S^{\ast}$-well-filteredness, with implications for semantics and topology of function spaces. The work advances understanding of how weakened filtration properties interact with standard constructions in domain theory.

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 ΣP 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.

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

TL;DR

The paper defines -well-filtered spaces as a strictly weaker analogue of strongly well-filtered spaces and analyzes their position relative to -spaces and weak well-filtered spaces. It proves that for any dcpo , the Scott space is a -space if and only if it is -well-filtered, and it exhibits Johnstone's non-sober dcpo as -well-filtered yet not strongly well-filtered, illustrating a clear separation of classes. The authors develop a broad suite of properties, including closure under subspaces, behavior under retracts, and nuanced product behavior; they also study the effect of Isbell-topologized function spaces and Smyth power spaces on -well-filteredness, establishing several transfer principles and counterexamples. Together, these results refine the hierarchy of domain-theoretic space classes and clarify how power-space constructions interact with -well-filteredness, with implications for semantics and topology of function spaces. The work advances understanding of how weakened filtration properties interact with standard constructions in domain theory.

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 ΣP 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.
Paper Structure (6 sections, 41 theorems, 8 equations, 2 figures)

This paper contains 6 sections, 41 theorems, 8 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The $d^{\ast}$-spaces
  4. $S^{\ast}$-well-filtered spaces
  5. $\mathrm{Some~basic~properties~of}~S^{\ast}$-well-filtered spaces
  6. $S^{\ast}$-well-filteredness of Smyth power spaces

Key Result

Lemma 2.8

(s2024) Let $\mathbf{K}$ be a full subcategory of $\mathbf{Top}_{0}$ with $\mathbf{K}\nsubseteq \mathbf{Top}_{1}$. Then $\mathbf{K}$ is reflective in $\mathbf{Top}_{0}$ if and only if $\mathbf{K}$ is productive and b-closed-hereditary.

Figures (2)

  • Figure 1: The poset $P$ in Example \ref{['e4']}.
  • Figure 2: The relationships of among spaces lying between $T_{2}$-space and weak well-filtered space.

Theorems & Definitions (72)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Lemma 2.8
  • Definition 3.1
  • Remark 3.2
  • ...and 62 more