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Preserving \(Z\)-sets by Dranishnikov’s resolution. (English) Zbl 1182.54040
A proper map \(h:Y\to Z\) between separable complete \(\text{ANE}(n)\) is a \(UV^{n-1}\)-divider of \(f:X\to Z\) if there exists a topological embedding \(g:X\to Y\) such that \(g(X)\) is dense in \(Y\), the inclusion of \(g(X)\) in \(Y\) is a \(UV^{n-1}\)-map and \(f=h\circ g\). Let \(Q\) be the Hilbert cube, \(\mu^k\) the \(k\)-dimensional Menger compactum and \(\nu^k\) the \(k\)-dimensional Nöbeling space. The authors prove that there exists a Dranishnikov resolution \(d_k:\mu^k\to Q\) and a Chigogidze resolution \(c_k:\nu^k\to Q\) such that \(d_k\) is a \(UV^{k-1}\)-divider of \(c_k\). This implies that \(d_k^{-1}(Z)\) is a Z-set in \(\mu^k\) for every Z-set \(Z\) in \(Q\).

MSC:
54F65 Topological characterizations of particular spaces
58B05 Homotopy and topological questions for infinite-dimensional manifolds
54C55 Absolute neighborhood extensor, absolute extensor, absolute neighborhood retract (ANR), absolute retract spaces (general properties)
57N20 Topology of infinite-dimensional manifolds
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