Effective Dispersion in Computable Metric Spaces
نویسنده
چکیده
We investigate the relationship between computable metric spaces (X, d, α) and (X, d, β), where (X, d) is a given metric space. In the case of Euclidean space, α and β are equivalent up to isometry, which does not hold in general. We introduce the notion of effectively dispersed metric space. This notion is essential in the proof of the main result of this paper: (X, d, α) is effectively totally bounded if and only if (X, d, β) is effectively totally bounded, i.e. the property that a computable metric space is effectively totally bounded (and in particular effectively compact) depends only on the underlying metric space.
منابع مشابه
Local Computability of Computable Metric Spaces and Computability of Co-c.e. Continua
We investigate conditions on a computable metric space under which each co-computably enumerable set satisfying certain topological properties must be computable. We examine the notion of local computability and show that the result by which in a computable metric space which has the effective covering property and compact closed balls each co-c.e. circularly chainable continuum which is not ch...
متن کاملThe Classification Problem for Compact Computable Metric Spaces
We adjust methods of computable model theory to effective analysis. We use index sets and infinitary logic to obtain classificationtype results for compact computable metric spaces. We show that every compact computable metric space can be uniquely described, up to an isomorphism, by a computable Π3 formula, and that orbits of elements are uniformly given by computable Π2 formulas. We show that...
متن کاملOn a metric generalization of the $tt$-degrees and effective dimension theory
In this article, we study an analogue of $tt$-reducibility for points in computable metric spaces. We characterize the notion of the metric $tt$-degree in the context of first-level Borel isomorphism. Then, we study this concept from the perspectives of effective topological dimension theory and of effective fractal dimension theory.
متن کاملEffective zero-dimensionality for computable metric spaces
We begin to study classical dimension theory from the computable analysis (TTE) point of view. For computable metric spaces, several effectivisations of zerodimensionality are shown to be equivalent. The part of this characterisation that concerns covering dimension extends to higher dimensions and to closed shrinkings of finite open covers. To deal with zero-dimensional subspaces uniformly, fo...
متن کاملComputable Banach Spaces via Domain Theory 1
This paper extends the domain-theoretic approach to computable analysis to complete metric spaces and Banach spaces. We employ the domain of formal balls to deene a computability theory for complete metric spaces. For Banach spaces, the domain specialises to the domain of closed balls, ordered by reversed inclusion. We characterise computable linear operators as those which map computable seque...
متن کامل