On Scott power spaces

TL;DR

This paper explores properties of Scott power spaces, showing how they preserve well-filteredness in certain spaces, examining sobriety conditions, and analyzing various topological properties of Smyth and Scott power spaces.

Contribution

It establishes new results on the preservation of well-filteredness and sobriety in Scott power spaces, and investigates their topological properties.

Findings

Scott power space of a well-filtered space remains well-filtered

A sober space can have a non-sober Scott power space

Conditions for sobriety of Scott power spaces

Abstract

In this paper, we mainly discuss some basic properties of Scott power spaces. For a $T_0$ space $X$, let $\mathsf{K}(X)$ be the poset of all nonempty compact saturated subsets of $X$ endowed with the Smyth order. It is proved that the Scott power space $\Sigma \mathsf{K}(X)$ of a well-filtered space $X$ is still well-filtered, and a $T_0$ space $Y$ is well-filtered iff $\Sigma \mathsf{K}(Y)$ is well-filtered and the upper Vietoris topology is coarser than the Scott topology on $\mathsf{K}(Y)$. A sober space is constructed for which its Scott power space is not sober. A few sufficient conditions are given under which a Scott power space is sober. Some other properties, such as local compactness, first-countability, Rudin property and well-filtered determinedness, of Smyth power spaces and Scott power spaces are also investigated.