Dimensionally valuable cell-like resolution for \(ANR\)-compacta (Q1311512)

It is shown that every ANR-compactum \(X\) has a cell-like resolution \(f: Y\to X\) by means of the ANR-compactum \(Y\) of dimension \(\dim Y=\dim X\) with Borsuk condition \((\Delta)\). The condition \((\Delta)\) on \(Y\) means the local contractibility of \(Y\) with the additional requirement that every small \(k\)-dimensional subset \(A\subset Y\) is contractible to a point in a small \((k+1)\)-dimensional subset \(B\subset Y\). Now it is known that there are ANR-compacta for which the Borsuk condition \((\Delta)\) does not hold.