Computability of Polish Spaces Up to Homeomorphism

Abstract

We study computable Polish spaces and Polish groups up to homeomorphism. We prove a natural effective analogy of Stone duality, and we also develop an effective definability technique which works up to homeomorphism. As an application, we show that there is a $\Delta ^0_2$ Polish space not homeomorphic to a computable one. We apply our techniques to build, for any computable ordinal $\alpha $, an effectively closed set not homeomorphic to any $0^{(\alpha )}$-computable Polish space; this answers a question of Nies. We also prove analogous results for compact Polish groups and locally path-connected spaces.

Computable topological abelian groups

“…For instance, we give the first known example of a ∆ 0 2 -metrized connected Polish space not homeomorphic to any computably metrized Polish space; note that we do not restrict ourselves to Polish groups here. This answers a question recently raised by Kihara and also independently posed in [25]. See §1.6.2 for further open questions and their solutions.…”

“…Kihara in his recent CiE2020 talk asked: (Q3) Is every ∆ 0 2 -metrized connected compact Polish space homeomorphic to a computably metrized one? This question was also independently raised in [25] (Question 2). Also, [25] (Question 3) asks the analogous question for ∆ 0 2 -metrized connected Polish groups.…”

“…For instance, the problem of comparing effectively compact and computably metrized presentation has recently attracted a considerable attention. It follows from [25,30] that every computably metrized Stone space is homeomorphic to an effectively compact one; see [25] for an explanation. In contrast, Lemma 3.21 of [30] gives an example of an effectively metrized compact Polish space which is not homeomorphic to an effectively compact Polish space.…”

“…This is similar to our question above stated for connected groups, but it is concerned with arbitrary connected Polish spaces. Somewhat unexpectedly, it takes much effort to prove that there exists a ∆ 0 2 -metrized Polish space not homeomorphic to a computably metrized one (answering a question posed by Selivanov [51]); see [30,25] for three substantially different proofs all of which are non-trivial. All known examples share the same feature, namely they use connected components of the constructed spaces to "code" an undecidable set.…”

“…It is well-known that, in general, 0 ′ can list all covers of a given compact computably metrized Polish space (see, e.g., [25,30] for the technical details). It follows that, without the assumption of effective compactness, the dual of a computably metrized connected G is ∆ 0 2 ; this is a significant improvement over the crude ∆ 1 1 bound.…”