标题： 可计算离散表示的空间 显示英文标题

标题： Computably discrete represented spaces

摘要： 在可计算拓扑中，如果一个表示空间的相等谓词是半可判定的，则称其为可计算离散的。虽然任何此类空间经典上都同构于自然数的一个初始段，但可计算离散表示空间的可计算同构类型展现出丰富的结构。 我们证明了广泛研究的可枚举可计算等价关系（ceers）类正好对应于可计算拟波兰可计算离散空间。我们利用可计算离散空间在可计算拓扑中构建了多个分离的例子。我们构造了一个没有可判定性质的可计算离散可计算拟波兰空间，一个可计算拟波兰可计算豪斯多夫空间且不能够存在从该空间到自然数的可计算单射，一个两个点的空间，它在可计算豪斯多夫意义下成立但不是可计算离散的，以及一个两个点的空间，它在可计算离散意义上成立但不是可计算豪斯多夫的。 我们还扩展了维拉布赫的一个例子，用于区分可计算正则空间与可计算正规空间。 显示英文摘要

摘要： In computable topology, a represented space is called computably discrete if its equality predicate is semidecidable. While any such space is classically isomorphic to an initial segment of the natural numbers, the computable-isomorphism types of computably discrete represented spaces exhibit a rich structure. We show that the widely studied class of computably enumerable equivalence relations (ceers) corresponds precisely to the computably Quasi-Polish computably discrete spaces. We employ computably discrete spaces to exhibit several separating examples in computable topology. We construct a computably discrete computably Quasi-Polish space admitting no decidable properties, a computably discrete and computably Hausdorff precomputably Quasi-Polish space admitting no computable injection into the natural numbers, a two-point space which is computably Hausdorff but not computably discrete, and a two-point space which is computably discrete but not computably Hausdorff. We further expand an example due to Weihrauch that separates computably regular spaces from computably normal spaces.