Several remarks on dimensions modulo ANR-compacta

We investigate a dimension function L - dim ( L is a class of ANR-compacta). Main results are as follows. be an ANR-compactum. L * L is not contractible, then for every n ⩾ 0 there is a cube I m with L - dim I m = n . L is simply connected and f : X → Y is an acyclic mapping from a finite-dimensional compact Hausdorff space X onto a finite-dimensional space Y, then L - dim Y ⩽ L - dim X . L is simply connected and L * L is not contractible, then for every n ⩾ 2 there exists a compact Hausdorff space Z n L such that L - dim Z n L = n , and for an arbitrary closed set F ⊂ Z n L either L - dim F ⩽ 0 or L - dim F = n .