Nöbeling spaces and pseudo-interiors of Menger compacta
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 \(\sigma ^{2n+1}_{n}=\mathbf R^{2n+1}-s^{2n+1}_{n}\).
scientific article; zbMATH DE number 858598
scientific article