We examine computability structures on a metric space and the relationships between maximal, separable and dense computability structures. We prove that in a computable metric space which has the effective covering property and compact closed balls for a given computable sequence which is a metric basis there exists a unique maximal computability structure which contains that sequence. Furthermore, we prove that each maximal computability structure on a convex subspace of Euclidean space is dense. We also examine subspaces of Euclidean space on which each dense maximal computability structure is separable and prove that spheres, boundaries of simplices and conics are such spaces.

我们研究了度量空间上的可计算性结构以及最大，可分离和密集的可计算性结构之间的关系。我们证明，在一个具有有效覆盖特性的可计算度量空间中，对于作为度量基础的给定可计算序列，它具有紧凑的闭合球，存在一个包含该序列的唯一最大可计算性结构。此外，我们证明了欧几里得空间的一个凸子空间上的每个最大可计算结构都是致密的。我们还检查了欧几里德空间的子空间，在该子空间上每个密集的最大可计算结构都是可分离的，并证明球体，单纯形的边界和圆锥形都是这样的空间。