Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Degree spectra of homeomorphism types of compact Polish spaces

Mathieu Hoyrup, Takayuki Kihara, Victor Selivanov

Abstract

A Polish space is not always homeomorphic to a computably presented Polish space. In this article, we examine degrees of non-computability of presenting homeomorphic copies of compact Polish spaces. We show that there exists a $0'$-computable low$_3$ compact Polish space which is not homeomorphic to a computable one, and that, for any natural number $n\geq 2$, there exists a Polish space $X_n$ such that exactly the high$_{n}$-degrees are required to present the homeomorphism type of $X_n$. We also show that no compact Polish space has a least presentation with respect to Turing reducibility. The first version of this article appeared in April 2020. A major update was made in September 2023, with improved proofs and results. This is the final version from January 2024, with more results on Čech homology groups.

Degree spectra of homeomorphism types of compact Polish spaces

Abstract

A Polish space is not always homeomorphic to a computably presented Polish space. In this article, we examine degrees of non-computability of presenting homeomorphic copies of compact Polish spaces. We show that there exists a -computable low compact Polish space which is not homeomorphic to a computable one, and that, for any natural number , there exists a Polish space such that exactly the high-degrees are required to present the homeomorphism type of . We also show that no compact Polish space has a least presentation with respect to Turing reducibility. The first version of this article appeared in April 2020. A major update was made in September 2023, with improved proofs and results. This is the final version from January 2024, with more results on Čech homology groups.

Paper Structure

This paper contains 30 sections, 39 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Presentations of Polish spaces
  4. Hyperspaces.
  5. Realizations
  6. Good covers
  7. Clopen subsets
  8. Covering dimension
  9. Cantor-Bendixson derivative
  10. Stone duality
  11. Countable spaces
  12. Computable aspects of Čech homology groups
  13. Background on Čech homology groups
  14. Simplicial complexes.
  15. Simplicial homology groups.
  16. ...and 15 more sections

Key Result

Lemma 2.1

Let ${X}$ be a compact Polish space. If ${X}$ has a $\mathbf{d}$-computable Polish presentation, then ${X}$ has a $\mathbf{d}'$-computable compact presentation.

Theorems & Definitions (88)

  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 78 more