Chigogidze, A. Ch. Compacta situated in an \(n\)-dimensional universal Menger compactum and having homeomorphic complements in it. (Russian) Zbl 0669.54010 Mat. Sb., N. Ser. 133(175), 481-496 (1987). The author proves among others the following nice theorem: Let \(\mu^{n+1}\) denote the \((n+1)\)-dimensional universal Menger compactum, \(n\geq 0\) and let compacta X and Y are Z-subsets of \(\mu^{n+1}\). Then the complements \(\mu^{n+1}\setminus X\) and \(\mu^{n+1}\setminus Y\) are homeomorphic iff \(n-Sh(X)=n-Sh(Y).\) The notion of n-shape of the compactum X, i.e. n-Sh(X) is the basic notion introduced by the author. It is a fruitful modificaton of the notion of ordinary shape introduced by K. Borsuk. The category of n-shape of compacta is obtained as the Borsuk’s shape category replacing the notion of ordinary homotopy by the following notion of n-homotopy: Let \(n\geq 0\). Continuous maps f,g: \(X\to Y\) of Polish spaces are called n- homotopic iff for every continuous map \(\alpha\) : \(Z\to X\) of a Polish space Z with dim \(Z\leq n\) the compositions \(f\alpha\) and \(g\alpha\) are homotopic in the ordinary sense. Reviewer: S.A.Antonyan Cited in 8 ReviewsCited in 7 Documents MSC: 54C56 Shape theory in general topology 55P55 Shape theory 54C55 Absolute neighborhood extensor, absolute extensor, absolute neighborhood retract (ANR), absolute retract spaces (general properties) Keywords:n-homotopy; universal Menger compactum; category of n-shape PDFBibTeX XMLCite \textit{A. Ch. Chigogidze}, Mat. Sb., Nov. Ser. 133(175), 481--496 (1987; Zbl 0669.54010) Full Text: EuDML