Extraordinary dimension of maps

We consider the extraordinary dimension dimL introduced recently by Shchepin [E.V. Shchepin, Arithmetic of dimension theory, Russian Math. Surveys 53 (5) (1998) 975–1069]. If L is a CW-complex and X a metrizable space, then dimLX is the smallest number n such that ΣnL is an absolute extensor for X, where ΣnL is the nth suspension of L. We also write dimLf⩽n, where is a given map, provided dimLf−1(y)⩽n for every y∈Y. The following result is established: Suppose is a perfect surjection between metrizable spaces, Y a C-space and L a countable CW-complex. Then conditions (1)–(3) below are equivalent:(1)dimLf⩽n;(2)There exists a dense and Gδ subset G of C(X,In) with the source limitation topology such that dimL(f×g)=0 for every g∈G;(3)There exists a map is such that dimL(f×g)=0;If, in addition, X is compact, then each of the above three conditions is equivalent to the following one;(4)There exists an Fσ set A⊂X such that dimLA⩽n−1 and the restriction map f|(X∖A) is of dimension dimf|(X∖A)⩽0.