A computable metric space is a triple (X,d,α), where (X,d) is a metric space and α = (αi) is a sequence in X whose image is dense in (X,d) and such that the function N2 → R, (i, j) 7→ d(αi,αj), is computable. If d is a complete metric, then we also say that (X,d,α) is a computable Polish space.
▷ We call the elements of the sequence (αi )i∈N the special points. We call such (αi )i∈N a computable structure of the metric space. Not necessarily unique.
We consider an abstract metric space with a computability structure and an effective separating set. In this article, we also introduce an effectively ...
Sep 25, 2023 · A structure A is computably presented if its domain is N and all the functions and relations on A are computable. This works very well for ...
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Jun 5, 2021 · We present a survey on computability of subsets of Euclidean space and, more generally, computability concepts on metric spaces and their subsets.
We examine computability structures on a metric space and the relationships between maximal, separable and dense computability structures.
Abstract: We investigate the relationship between computable metric spaces (X, d, α) and (X, d, β), where (X, d) is a given metric space.
Computability theoretic aspects of Polish metric spaces are studied by adapting notions and methods of computable structure theory.
We say that an uncountable metric space is computably categor- ical if every two computable structures on this space are equivalent up to a computable isometry.
Introduction · Computability structures · The main question · A known result for sub-spaces of the Euclidean space · Main result for more general metric spaces.