Computability of a topological poset
type: Item
instance of: scholarly article
title: Computability of a topological poset
zbMATH Open document ID: 1090.54032
author: W. D. Gillam
published in: Topology and its Applications
publication date: 2006-03-20T00:00:00Z
review text: In the paper [Topology Appl. 148, 63--82 (2005; Zbl 1062.54039)], the author of the paper under review constructed to a set \(\mathbb{Q}\) the poset \({\mathcal P}(\mathbb{Q})/{\sim}= \{\overline{X}=\{Y\subseteq \mathbb{Q}:\;Y\sim X\}\}\). For this poset the partial order is defined as follows: \(\overline{X}\leq_h\overline{Y}\) if \(X\leq_hY\), where for subspaces \(X,\,Y\subseteq\mathbb{Q}\), \(X\leq_hY\) means that \(X\) is homeomorphic to a subspace of \(Y\) and \(X\sim Y\) means \(X\leq_hY\leq_hX\). In [loc. cit.], it was established that \({\mathcal P}(\mathbb{Q})/{\sim}\) is essentially determined by considering only the scattered subset \(X\subseteq \mathbb{Q}\) of finite Cantor-Bendixson rank \(N(X)\). For the purpose of the present paper the author considers only scattered countable metric spaces with finite \(N(X)\) and shows that the set \(\mathbb{A}=\{X\subseteq \mathbb{Q}:\;1<N(X)<\omega\}/{\sim}\) can be generated by a computer program. As an example, a complete list of homeomorphism types of spaces of rank \(\leq 4\) and a presentation of the embeddability ordering of these spaces is given.
reviewed by: Badri P. Dvalishvili
Mathematics Subject Classification ID: 54H10
Mathematics Subject Classification ID: 54H12
Mathematics Subject Classification ID: 54D35
Mathematics Subject Classification ID: 06A06
zbMATH DE Number: 5013529
zbMATH Keywords: embedding
zbMATH Keywords: partially-ordered
zbMATH Keywords: scattered
zbMATH Keywords: metric space
zbMATH Keywords: computable
zbMATH Keywords: rank
zbMATH Keywords: type of a point
zbMATH Keywords: homeomorphism type of a space
MaRDI profile type: Publication
full work available at URL: j.topol.2005.03.004
OpenAlex ID: W2003454604
cites work: Embeddability properties of countable metric spaces
DOI: 10.1016/J.TOPOL.2005.03.004
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description: scientific article; zbMATH DE number 5013529