The approximate section extension property and hereditary shape equivalences (Q1056955)

The results can be regarded as shape versions of some of the results in the classical paper by \textit{A. Dold} [Ann. Math., II. Ser. 78, 223-255 (1963; Zbl 0203.254)] on the homotopy theory of fibrations. The author introduces the concept of the approximate section extension property (ASEP). This is complicated to define, but roughly stated, a map \(f: X\to Y\) between metric spaces is said to have the ASEP, if whenever f is suitably embedded into a map \(F: M\to N\) between ANR's, then approximate local sections of f over closed subsets in Y can be extended to approximate sections of f over Y. Here approximate is expressed in terms of open neighbourhoods of X in M and open coverings of N. As the main results it is proved that hereditary shape equivalences are exactly the maps with the hereditary ASEP, and that every UV\({}^{n-1}\)-map with an n-dimensional range and every UV\({}^{\infty}\)-map between ANR's has the ASEP.