Extraordinary dimension of maps

We consider the extraordinary dimension dim L 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 dim L X is the smallest number n such that Σ n L is an absolute extensor for X, where Σ n L is the nth suspension of L. We also write dim L f ⩽ n , where f : X → Y is a given map, provided dim L f −1 ( y ) ⩽ n for every y ∈ Y . The following result is established: Suppose f : X → Y is a perfect surjection between metrizable spaces, Y a C-space and L a countable CW-complex. Then conditions (1)–(3) below are equivalent: f ⩽ n ; exists a dense and G δ subset G of C ( X , I n ) with the source limitation topology such that dim L ( f × g ) = 0 for every g ∈ G ; exists a map g : X → I n is such that dim L ( f × g ) = 0 ; addition, X is compact, then each of the above three conditions is equivalent to the following one; exists an F σ set A ⊂ X such that dim L A ⩽ n − 1 and the restriction map f | ( X ∖ A ) is of dimension dim f | ( X ∖ A ) ⩽ 0 .