A universal separable metric space based on the triangular Sierpiński curve
type: Item
instance of: scholarly article
title: A universal separable metric space based on the triangular Sierpiński curve
zbMATH Open document ID: 1003.54019
DOI: 10.1016/S0166-8641(01)00017-7
author: Ivan Ivanšić
author: Uroš Milutinović
published in: Topology and its Applications
publication date: 2002-06-27T00:00:00Z
review text: The authors construct, for each \(n\), a space \(\Sigma(3)^{n+1}\) out of copies of the triangular Sierpiński curve \(\Sigma(3)\); co-author Milutinović has shown in earlier work that \(\Sigma(3)\) is homeomorphic to a construction of S. L. Lipscomb. They go on to define \(L_n(3)= \{x\in\Sigma(3)^{n+1}\): at least one coordinate of \(x\) is irrational\}, where an irrational point of \(\Sigma(3)\) is one that is not a vertex of one of the triangles formed in the construction of \(\Sigma(3)\). The authors then present a proof that this space \(L_n(3)\) is a universal space for all separable \(n\)-dimensional metric spaces. They go on to announce that the same methods will obtain the following Theorem (the details will appear elsewhere): Let \(X\) be any separable \(n\)-dimensional metric space, and \(f: X\to \Sigma(3)^{n+1}\) any continuous mapping. Then for any \(\varepsilon> 0\) there is an embedding \(\psi: X\to L_n(3)\subseteq \Sigma(3)^{n+1}\) such that \(d(f,\psi)\leq \varepsilon\).
Mathematics Subject Classification ID: 54F45
zbMATH DE Number: 1761006
zbMATH Keywords: decompositions
zbMATH Keywords: universal space
zbMATH Keywords: separable \(n\)-dimensional metric space
zbMATH Keywords: embedding
reviewed by: Bruce R. Wenner
MaRDI profile type: Publication
cites work: Q4776600
cites work: On Imbedding Finite-Dimensional Metric Spaces
cites work: Lipscomb's L(A) Space Fractalized in Hilbert's l 2 (A) Space
cites work: Q3136012
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description: scientific article; zbMATH DE number 1761006