The surjection property and computable type
type: Item
instance of: scholarly article
title: The surjection property and computable type
zbMATH Open document ID: 1548.03044
author: Q6062175
author: Mathieu Hoyrup
published in: Topology and its Applications
publication date: 2024-08-23T00:00:00Z
review text: The authors provide a detailed, further study of two properties of spaces and pairs of spaces, the surjection property and the \(\varepsilon\)-surjection property. These were introduced by them in [LIPIcs -- Leibniz Int. Proc. Inform. 229, Article 111, 16 p. (2022; Zbl 07870321)] to characterize the notion of computable type arising from computability theory. They state that, at least informally, a metrizable compactum has \textit{computable type} if for every embedded copy \(X\) of it in the Hilbert cube, if the set of metric balls that are disjoint from \(X\) can be enumerated by a Turing machine, then the set of metric balls intersecting \(X\) can be enumerated by a Turing machine. One would have to look further into the references, however, to find a complete definition of this notion. Let us review a definition from Section 2.\N\NDefinition 2.1. A pair \((X,A)\) has the \textit{surjection property} if every continuous function \(f:X\to X\) such that \(f|A\) is the identity is surjective. If \((X,d)\) is a metric space, \(A\subset X\) is a closed subset, and for all \(\varepsilon>0\), every continuous function \(f:X\to X\) with \(f|A\) equal the identity and \(d(f,\mathrm{id}_X)<\varepsilon\) is a surjection, then we say that \((X,A)\) has the \(\varepsilon\)-\textit{surjection property}. In the special case that \(A=\emptyset\), then one says that \(X\) has the \(\varepsilon\)-\textit{surjection property}.\N\NThe focus of this research is on studying the properties given in Definition 2.1. and in relating them to computability theory. Although this definition is given in a general form, its application will be to compact metric spaces \(X\) and closed, hence compact, subspaces \(A\subset X\). The organization of the paper goes as follows. Section 2 contains an investigation of general aspects of the \(\varepsilon\)-surjection property, such as its preservation under countable unions. The next section includes a result showing that for finite unions of cones, th…
reviewed by: Leonard R. Rubin
Mathematics Subject Classification ID: 03D78
Mathematics Subject Classification ID: 57Q05
Mathematics Subject Classification ID: 55U10
zbMATH DE Number: 7900744
zbMATH Keywords: simplicial complex
zbMATH Keywords: computable compact set
zbMATH Keywords: surjection property
zbMATH Keywords: cone space
zbMATH Keywords: maps to spheres
zbMATH Keywords: null-homotopy
MaRDI profile type: Publication
cites work: Q5089548
cites work: Effectiveness for Embedded Spheres and Balls
cites work: Compact manifolds with computable boundaries
cites work: Computability of products of chainable continua
cites work: Are two given maps homotopic? An algorithmic viewpoint
cites work: Q5532996
cites work: Q4819371
cites work: Q5837247
cites work: On products in homotopy groups
cites work: Q5815345
cites work: Q5768015
cites work: A generalization of the Hopf invariant
cites work: On computable numbers, with an application to the Entscheidungsproblem.
cites work: Q3639062
cites work: Computability of 1-manifolds
cites work: Semicomputable manifolds in computable topological spaces
cites work: Warsaw discs and semicomputability
cites work: Chainable and circularly chainable semicomputable sets in computable topological spaces
cites work: Computability of graphs
cites work: Computability of pseudo-cubes
cites work: The infinite-dimensional topology of function spaces
cites work: Über eine Klasse von lokal zusammenhängenden Räumen
cites work: Note on a theorem due to Borsuk
cites work: Q5200638
DOI: 10.1016/J.TOPOL.2024.109020
description: scientific article; zbMATH DE number 7900744