On $\mathbf{K}$-reflections of Scott spaces

TL;DR

This paper explores conditions under which Scott spaces retain their properties after $ extbf{K}$-reflection and investigates the existence of Scott $ extbf{K}$-completions of posets, providing necessary and sufficient criteria.

Contribution

It establishes necessary and sufficient conditions for the $ extbf{K}$-reflection of Scott spaces to remain Scott spaces and for Scott $ extbf{K}$-completions of posets to exist.

Findings

Neither sobrification nor well-filtered reflection of Johnstone space is Scott.

Conditions for $ extbf{K}$-reflection of Scott spaces to be Scott spaces.

Discussion on $ extbf{K}$-reflections of Alexandroff spaces.

Abstract

In this paper, for a full subcategory $\mathbf{K}$ of the category of all $T_0$ spaces with continuous mappings, we investigate the questions under what conditions the $\mathbf{K}$-reflection of a Scott space is still a Scott space and under what conditions the Scott $\mathbf{K}$-completion of a poset exists. Some necessary and sufficient conditions for the $\mathbf{K}$-reflection of a Scott space to be a Scott space and for the existence of Scott $\mathbf{K}$-completion of a poset are established, respectively. It is shown that neither the sobrification nor the well-filtered reflection of the Johnstone space is a Scott space. The $\mathbf{K}$-reflections of Alexandroff spaces and the $\mathbf{K}$-completions of posets are also discussed.