Work

Computable Topological Presentations

Year: 2026

Type: article

Abstract: A computable topological presentation of a space is given by an effective list of a countable basis of non-empty open sets so that the intersection of the basic sets is uniformly effectively enumerable. We show that every countably-based $T_0$ -space has a computable topological presentation, and that, conversely, every (formal) computable topological presentation represents some Polish space. In the compact case, we give a computable uniform list of computable topological presentations such that every compact Polish space is represented by exactly one presentation from the list. Note that none of these results assume that the Polish (or $T_0$ ) spaces are effective. Quite surprisingly, the effectively compact topological presentations turn out to be rather well behaved. Not only do such presentations allow one to construct a $\Delta ^0_2$ (complete) metric compatible with the topology, but also, under a mild extra condition, they can be turned into a computably compact Polish presentation of the space.

Source: Journal of Symbolic Logic

DOI: 10.1017/jsl.2026.10190

Authors: ALEXANDER MELNIKOV, KENG MENG NG, MATHIEU HOYRUP

Institutions: Victoria University of Wellington, MOCQUA: Modèles de calcul émergents, Nanyang Technological University, Institut national de recherche en sciences et technologies du numérique

Language: English

Topic: Computability, Logic, AI Algorithms

Subfield: Computational Theory and Mathematics

Field: Computer Science

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,550

OpenAlex ID: W4405659265

FWCI:

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