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Separating Notions in Effective Topology

INTERNATIONAL JOURNAL OF ALGEBRA AND COMPUTATION(2023)

Victoria Univ Wellington | Nanyang Technol Univ

Cited 0|Views10
Abstract
We compare several natural notions of effective presentability of a topological space up to homeomorphism. We note that every left-c.e. (lower-semicomputable) Stone space is homeomorphic to a computable one. In contrast, we produce an example of a locally compact, left-c.e. space that is not homeomorphic to any computable Polish space. We apply a similar technique to produce examples of computable topological spaces not homeomorphic to any right-c.e. (upper-semicomputable) Polish space, and indeed to any arithmetical or even analytical Polish space. We then apply our techniques to totally disconnected locally compact (tdlc) groups. We prove that every computably locally compact tdlc group is topologically isomorphic to a computable tdlc group.
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Computable topology,computable topological groups
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要点】:本文研究了不同类型的有效拓扑表示,发现了左可计算(lower-semicomputable)Stone空间与可计算空间的同胚关系,并构造了特定类型的非同胚空间,同时探讨了计算局部紧致完全断续群的同构问题。

方法】:通过对比分析不同的有效表示概念,利用构造性方法来展示不同类型空间之间的同胚与非同胚关系。

实验】:未明确提到具体实验,但通过理论构造展示了左可计算Stone空间与可计算空间的同胚性,以及非同胚的例子,如局部紧致、左可计算空间与可计算波兰空间的非同胚性。同时,对计算局部紧致完全断续群进行了同构性证明。未提及具体的数据集名称。