论文标题
强大的计算类型
Strong computable type
论文作者
论文摘要
紧凑的集合具有可计算类型的类型，如果该集合的任何同构副本可被计算。米勒证明，有限维球具有可计算类型的Iljazović，而其他作者为许多其他集合（例如歧管）建立了该属性。在本文中，我们提出了一项关于可计算类型概念的理论研究，以提高我们对这一概念的一般理解并提供证明或反驳这一属性的工具。 我们首先表明，在文献中区分的可计算类型的定义分别涉及度量空间和Hausdorff空间，实际上是等效的。我们认为，可计算类型的更强大的相关版本更具举止，并且容易出现拓扑分析。我们获得了与拓扑不变性的描述性复杂性以及纯粹的拓扑标准有关的强大计算类型的特征。我们研究了低描述性复杂性的拓扑不变性的两个家族，表达了连续函数的可扩展性和无效的态度。我们将理论重新审视以前的结果并获得新的结果。
A compact set has computable type if any homeomorphic copy of the set which is semicomputable is actually computable. Miller proved that finite-dimensional spheres have computable type, Iljazović and other authors established the property for many other sets, such as manifolds. In this article we propose a theoretical study of the notion of computable type, in order to improve our general understanding of this notion and to provide tools to prove or disprove this property. We first show that the definitions of computable type that were distinguished in the literature, involving metric spaces and Hausdorff spaces respectively, are actually equivalent. We argue that the stronger, relativized version of computable type, is better behaved and prone to topological analysis. We obtain characterizations of strong computable type, related to the descriptive complexity of topological invariants, as well as purely topological criteria. We study two families of topological invariants of low descriptive complexity, expressing the extensibility and the null-homotopy of continuous functions. We apply the theory to revisit previous results and obtain new ones.
