Topological aspects of poset spaces

TL;DR

This paper explores the topological properties of poset spaces formed by filters, providing a complete characterization of countably based MF spaces and applying these results to domain theory.

Contribution

It offers a thorough analysis of MF and UF spaces, characterizes countably based MF spaces, and connects these findings to domain representations in topology.

Findings

Countably based MF spaces are exactly second-countable T_1 spaces with the strong Choquet property.

Characterization of second-countable spaces with domain representations.

Topological properties of filter-based poset spaces are thoroughly described.

Abstract

We study two classes of spaces whose points are filters on partially ordered sets. Points in MF spaces are maximal filters, while points in UF spaces are unbounded filters. We give a thorough account of the topological properties of these spaces. We obtain a complete characterization of the class of countably based MF spaces: they are precisely the second-countable T_1 spaces with the strong Choquet property. We apply this characterization to domain theory to characterize the class of second-countable spaces with a domain representation.