# Domain-complete and LCS-complete spaces

## TL;DR

This paper introduces and studies domain-complete and LCS-complete spaces, broadening the understanding of their topological properties and relationships to known classes like Polish and quasi-Polish spaces.

## Contribution

It defines new classes of spaces, establishes their properties, and characterizes their relationships to existing topological spaces, including applications to valuations and measures.

## Key findings

- LCS-complete spaces are sober, Wilker, and consonant.
- Countably-based LCS-complete spaces are exactly the quasi-Polish spaces.
- Metrizable LCS-complete spaces are the completely metrizable spaces.

## Abstract

We study $G_\delta$ subspaces of continuous dcpos, which we call domain-complete spaces, and $G_\delta$ subspaces of locally compact sober spaces, which we call LCS-complete spaces. Those include all locally compact sober spaces-in particular, all continuous dcpos-, all topologically complete spaces in the sense of \v{C}ech, and all quasi-Polish spaces-in particular, all Polish spaces. We show that LCS-complete spaces are sober, Wilker, compactly Choquet-complete, completely Baire, and $\odot$-consonant-in particular, consonant; that the countably-based LCS-complete (resp., domain-complete) spaces are the quasi-Polish spaces exactly; and that the metrizable LCS-complete (resp., domain-complete) spaces are the completely metrizable spaces. We include two applications: on LCS-complete spaces, all continuous valuations extend to measures, and sublinear previsions form a space homeomorphic to the convex Hoare powerdomain of the space of continuous valuations.