Constructing Compacta from Posets

TL;DR

This paper introduces a straightforward method to construct second countable T_1 compact spaces from countable posets with finite levels, enabling a duality approach similar to classical open cover constructions.

Contribution

It presents a novel, simple construction technique for compacta from posets, providing an abstract duality framework for building such spaces from finite components.

Findings

Applicable to all second countable T_1 compacta

Enables construction from finite building blocks

Provides an abstract analog of classical open cover methods

Abstract

We develop a simple method of constructing topological spaces from countable posets with finite levels, one which applies to all second countable T_1 compacta. This results in a duality amenable to building such spaces from finite building blocks, essentially an abstract analog of classical constructions defining compacta from progressively finer open covers.