Nöbeling spaces and pseudo-interiors of Menger compacta
type: Item
instance of: scholarly article
title: Nöbeling spaces and pseudo-interiors of Menger compacta
zbMATH Open document ID: 0869.57024
DOI: 10.1016/0166-8641(96)00041-7
published in: Topology and its Applications
publication date: 1997-08-28T00:00:00Z
review text: This interesting paper deals with the well-known problem of characterizing the universal Nöbeling space \(\mathbf N^{2n+1}_{n}\) as a space \(X\) satisfying the following properties: (a) \(X\) is separable and completely metrizable; (b) \(\text{dim} X=n\); (c) \(X\) is \(LC^{n-1}\cap C^{n-1}\); (d) any map of any at most \(n\)-dimensional, separable, completely metrizable space into \(X\) can be arbitrarily closely approximated by closed embeddings. The authors note five known constructions of spaces with the properties (a)-(d). One of them is the \(n\)-dimensional pseudo-interior \(s^{2n+1}_{n}\) of \(\mathbf R^{2n+1}\) [\textit{R. Geoghegan} and \textit{R. R. Summerhill}, Trans. Am. Math. Soc. 194, 141-165 (1974; Zbl 0288.57001)], another one is the pseudo-interior \(\nu ^{n}\) of the \(n\)-dimensional Menger compacta \(\mu ^{n}\) [the first author, Russ. Math. Surv. 44, No. 5, 145-174 (1989); translation from Usp. Mat. Nauk 44, No. 5(269), 117-140 (1989; Zbl 0696.54013)]. The paper is organized as follows. In Section 2 some preliminary facts are given, Section 3 contains a construction of the \(n\)-dimensional pseudo-interior \(\nu (k,n)\) and pseudo-boundary \(\Sigma (k,n)=\mu ^{k}-\nu (k,n)\) of \(\mu ^{k}\) for each \(k\geq n\). When \(k=n\), \(\nu ^{n}=\nu (n,n)\) and \(\Sigma (n,n)\) is denoted by \(\Sigma ^{n}\). In Section 4 the authors introduce the notion of the geometric pseudo-boundary and pseudo-interior of \(\mu ^{n}\) based on the Menger construction. It is proved that \(\nu ^{n}\) is homeomorphic to the geometric pseudo-interior (Section 5) and to the Nöbeling space \(\mathbf N^{k}_{n}\) (\(k\geq 2n+1\)) (Section 6). Section 7 contains some applications of the main results, in particular that every Polish absolute extensor is an \(n\)-soft image of \(\mathbf N^{2n+1}_{n}\) and that \(\mathbf N^{2n+1}_{n}\) is homeomorphic to \(s^{2n+1}_{n}\). It is also proved that \(\Sigma ^{n}\) is homeomorphic to the universal pseudo-boundary \(\sigma ^{2n+1}_{n…
Mathematics Subject Classification ID: 57Q12
Mathematics Subject Classification ID: 55P15
Mathematics Subject Classification ID: 54F45
Mathematics Subject Classification ID: 54F35
zbMATH DE Number: 858598
zbMATH Keywords: universal Nöbeling space
zbMATH Keywords: closed embeddings
zbMATH Keywords: pseudo-boundary
zbMATH Keywords: pseudo-interior
zbMATH Keywords: Menger construction
author: Alex Chigogidze
author: Kazuhiro Kawamura
author: E. D. Tymchatyn
reviewed by: V. Valov
MaRDI profile type: Publication
full work available at URL: 0166-8641(96)00041-7
OpenAlex ID: W1979728719
cites work: Q4060144
cites work: Characterizing 𝑘-dimensional universal Menger compacta
cites work: Characterizing certain incomplete infinite-dimensional absolute retracts
cites work: Dense embeddings of nowhere locally compact separable metric spaces
cites work: Limitation Topologies on Function Spaces
cites work: General Position Properties Satisfied by Finite Products of Dendrites
cites work: Q3308704
cites work: \(n\)-soft mappings of \(n\)-dimensional spaces
cites work: The theory ofn-shapes
cites work: \(UV^ n\)-equivalence and \(n\)-equivalence
cites work: NONCOMPACT ABSOLUTE EXTENSORS IN DIMENSIONn,n-SOFT MAPPINGS, AND THEIR APPLICATIONS
cites work: Classification Theorem for Menger Manifolds
cites work: Q3996849
cites work: Universal Maps and Surjective Characterizations of Completely Metrizable LC n -Spaces
cites work: k-dimensional skeletoids in \({\mathbb{R}}^ n\) and the Hilbert cube
cites work: Q3331924
cites work: Classification of Finite-Dimensional Universal Pseudo-Boundaries and Pseudo-Interiors
cites work: Q3725165
cites work: UNIVERSAL MENGER COMPACTA AND UNIVERSAL MAPPINGS
cites work: Q4087996
cites work: Dimension preserving compactifications allowing extensions of continuous functions
cites work: Menger spaces and inverse limits
cites work: Pseudo-Boundaries and Pseudo-Interiors in Euclidean Spaces and Topological Manifolds
cites work: Q5509701
cites work: Characterizing the Topology of Infinite-Dimensional σ-Compact Manifolds
cites work: Soft maps of manifolds
cites work: THE EMBEDDING OF COMPACTA IN EUCLIDEAN SPACE
cites work: Q5649665
cites work: Characterizing Hilbert space topology
cites work: The ambient homeomorphy of an incomplete subspace of infinite-dimensional Hilbert spaces
cites work: Q3998752
description: scientific article; zbMATH DE number 858598