When do weakly first-countable spaces and the Scott topology of open set lattice become sober?

TL;DR

The paper addresses when sober behavior arises for weakly first-countable spaces via the Scott topology on open-set lattices. It develops Fréchet-space techniques to align product Scott topologies with ordinary product topologies, establishing that $Σ(P×Q)$ equals $ΣP×ΣQ$ under the Fréchet condition and deriving sobriety results for related spaces. A key result shows that if $X$ is an $ω$-well-filtered coherent $d$-space with $X×X$ Fréchet, then $X$ is sober, broadening sobriety criteria beyond first-countable cases. The authors further prove sobriety of $Σ\mathcal{O}(X)$ for $ω$-type P-spaces and consonant Wilker spaces, and examine the sobriety of open-set lattices in various settings, including countable posets and $ ext{σ}$-constructions. Finally, they provide a counterexample demonstrating that sobriety and $T_{1}$ do not guarantee Hausdorffness in countable spaces.

Abstract

In this paper, we investigate the sobriety of weakly first-countable spaces and give some sufficient conditions that the Scott topologies of the open set lattices are sober. The main results are: (1) Let $P$ and $Q$ be two posets. If $ΣP\times ΣQ$ is a Fréchet space, then $Σ(P\times Q)=ΣP \times ΣQ$. (2) For every $ω$-well-filtered coherent $d$-space $X$, if $X\times X$ is a Fréchet space, then $X$ is sober; (3) For every $ω$ type P-space or consonant Wilker space $X$, $Σ\mathcal{O}(X)$ is sober.

Theorems & Definitions (42)

• Definition 3.1
• Example 3.2
• Theorem 3.3
• Theorem 3.4
• proof
• Theorem 3.5
• Corollary 3.6
• Corollary 3.7
• Corollary 3.8
• Theorem 3.9