Isometries and Computability Structures
Isometries and Computability Structures
复制标题
等轴测和可计算结构
DOI：
10.3217/jucs-016-18-2569
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发表时间：
2010
期刊：
影响因子：
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通讯作者：
Zvonko Iljazović
中科院分区：
文献类型：
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作者：
Zvonko Iljazović
我们研究了可计算度量空间（X, d, α）和（X, d, β）之间的关系，其中（X, d）是一个给定的度量空间。在欧几里得空间中，和在等距上是等价的，这在一般情况下是不成立的。我们引入了有效分散度量空间的概念，并将其用于证明以下结果：如果（X, d,）是有效全有界的，那么（X, d,）也是有效全有界的。这意味着可计算度量空间是有效完全有界的（特别是有效紧化的）这一性质仅取决于底层度量空间。在本文的最后一节，我们研究紧致度量空间（X, d），使得只有有限个等距X->X。在这种情况下，我们证明了一个比前一个更强的结果：如果（X, d, α）是有效完全有界的，那么α和β是等价的。因此，如果（X, d,）是有效完全有界的，则（X, d）具有唯一的可计算结构。
We investigate the relationship between computable metric spaces (X, d, alpha) and (X, d, beta), where (X, d) is a given metric space. In the case of Euclidean space, alpha and beta are equivalent up to isometry, which does not hold in general. We introduce the notion of effectively dispersed metric space and we use it in the proof of the following result: if (X, d, alpha) is effectively totally bounded, then (X, d, beta) is also effectively totally bounded. This means that the property that a computable metric space is effectively totally bounded (and in particular effectively compact) depends only on the underlying metric space. In the final section of this paper we examine compact metric spaces (X, d) such that there are only finitely many isometries X->X. We prove that in this case a stronger result holds than the previous one: if (X, d, alpha) is effectively totally bounded, then alpha and beta are equivalent. Hence if (X, d, alpha) is effectively totally bounded, then (X, d) has a unique computability structure.