Strong well-filteredness of upper topology on sup-complete posets

TL;DR

This paper introduces strong R-spaces, a new class of $T_0$ spaces, and proves that the upper topology on sup-complete posets and Hoare power spaces are strongly well-filtered, resolving recent open problems.

Contribution

It defines strong R-spaces and demonstrates that upper topologies on sup-complete posets are strongly well-filtered, addressing open questions in the field.

Findings

Upper topology on sup-complete posets is strongly well-filtered.

Hoare power space of a $T_0$-space is strongly well-filtered.

Introduces and investigates the new class of strong R-spaces.

Abstract

We first introduce and investigate a new class of $T_0$ spaces -- strong R-spaces, which are stronger than both R-spaces and strongly well-filtered spaces. It is proved that any sup-complete poset equipped with the upper topology is a strong R-space and the Hoare power space of a $T_0$-space is a strong R-space. Hence the upper topology on a sup-complete poset is strongly well-filtered and the Hoare power space of a $T_0$-space is strongly well-filtered, which answers two problems recently posed by Xu.