An effective version of the Stone duality

TL;DR

This paper develops an effective version of Stone duality for certain topological spaces and algebraic structures, integrating computability theory with topology to establish new dualities and realizations.

Contribution

It introduces effective notions of topological spaces and proves an effective Stone duality, extending computability results to topological and algebraic structures.

Findings

Effective duality between almost semispectral spaces and distributive c-posets.

Realization of any degree spectrum of a countable algebraic structure as a topological space.

Existence of computable topological spaces with a fixed number of computable copies.

Abstract

The paper studies computability-theoretic aspects of topological $T_0$-spaces. We introduce effective versions of the notions of a countable $c$-poset and a (second-countable) topological space with base. Based on this, we prove an effective version of the known Stone-type duality between the category $\mathbf{AS}$ (whose objects are almost semispectral spaces with base and whose morphisms are spectral mappings) and the category $\mathbf{DP}$ (whose objects are distributive $c$-posets and whose morphisms are strict mappings). Namely, we show that for an arbitrary set $Z\subseteq \omega$, this duality is preserved when one restricts to objects which have $Z$-computably enumerable presentations only. Following this approach, we establish several results in computable topology. We prove that every degree spectrum of a countable algebraic structure can be realized as the degree spectrum of a topological space with base. We show that for any non-zero natural number $N$, there is a computable topological space with base that has precisely $N$-many computable copies, up to effective spectral homeomorphisms.