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Compacta situated in an \(n\)-dimensional universal Menger compactum and having homeomorphic complements in it. (Russian) Zbl 0669.54010

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

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)
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