\(n\)-soft mappings of \(n\)-dimensional spaces (Q1173941)

The author improves on his results announced in [Sov. Math., Dokl. 30, 342-345 (1984); translation from Dokl. Akad. Nauk SSSR, 278, No. 1, 50-53 (1984; Zbl 0596.54012), and Izv. Akad. Nauk. SSSR, Ser. Mat. 50, No. 1, 156-180 (1986; Zbl 0603.54018)]. The author proves the following lemmas: 1) Suppose that \(n\in \omega\). For every countable locally finite simplicial complex \(K\), there exist an at most \(n\)-dimensional Polish space \(A^ n(K)\) and an \(n\)-soft mapping \(f_ K^ n: A^ n(K)\to | K|\). 2) Suppose that \(n\in \omega\). Every Polish space is an \(n\)-soft image of an at most \(n\)-dimensional Polish space. 3) Suppose that \(n\in \omega\). If all adjacent projections in the spectrum \(S=\{X_ k,\;p_ k^{k+1},\omega\}\) formed by Polish spaces are \(n\)-soft and \(n\)-filled, then the limit projection \(p_ 0:\;\lim S\to X\) is strongly \(n\)-universal. He uses these lemmas in the proof of the Theorem: For any \(n\in \omega\) and any Polish space \(X\) there exists an \(n\)-dimensional Polish space \(V^ n(X)\) and an \(n\)-soft mapping \(f_ X^ n: V^ n(X)\to X\) that possesses the following strong \(n\)-universality property. For any open covering \(U\) of \(V^ n(X)\) and any mapping \(g: Y\to V^ n(X)\) of an arbitrary at most \(n\)-dimensional Polish space \(Y\) into \(V^ n(X)\) there exists a closed embedding \(h:\;Y\to V^ n(X)\), that is \(U\)-close to \(g\) such that \(f_ X^ nh=f_ X^ ng\).