\(UV^ n\)-equivalence and \(n\)-equivalence
The following problem is considered: Suppose \(X\) and \(Y\) are two compacta embedded in the Hilbert cube \(Q\) as \(Z\)-sets. Can the property ``\(X\) and \(Y\) are \(\text{UV}^ n\)-equivalent (or \(n\)-equivalent)'' be described in terms of their complements \(Q-X\) and \(Q-Y\)? To solve this problem, the author first for every \(n\geq 0\) constructed an \((n+1)\)-invertible \(\text{UV}^ n\)-surjection \(f^ n: M^{n+1}\to Q\) from the \((n+1)\)- dimensional Menger compactum \(M^{n+1}\) onto \(Q\) satisfying the properties: (i) for each \((n+1)\)-dimensional compactum \(Z\), each open cover \(\gamma\) of \(M^{n+1}\) and each map \(h: Z\to M^{n+1}\) there is a fibered (with respect to \(f^ n\)) \(Z_{n+1}\)-embedding \(g: Z\to M^{n+1}\) which is \(\gamma\)-close to \(h\) and such that \(f^ n\circ g=f^ n\circ h\); (ii) there is a subspace \(A\subset M^{n+1}\) such that \(f^ n\mid A: A\to Q\) is an \((n+1)\)-soft map and \(M^{n+1}-A\) is \(\sigma Z\)-set in \(M^{n+1}\); (iii) if \(Z\) is a \(Z_{n+1}\)-set in \(Q\) then \((f^ n)^{-1}(Z)\) is a \(Z\)-set in \(M^{n+1}\). Then using the maps \(f^ n\) the author proved that if \(X\) and \(Y\) are \(Z\)-sets in \(Q\) we have \(X\) and \(Y\) are \((n+1)\)-equivalent if and only if \(Q-X\) and \(Q-Y\) are properly \(\text{UV}^ n\)-equivalent. If in addition \(X\) and \(Y\) are \(\text{LC}^ n\) then the last two conditions are equivalent to the following one: \(X\) and \(Y\) are \(\text{UV}^ n\)-equivalent.
scientific article; zbMATH DE number 90164
scientific article