Approximation of the Hausdorff distance by the distance of continuous surjections

The aim of this paper is to answer the following question: let (X,ϱ)(X,ϱ) and (Y,d)(Y,d) be metric spaces, let A,B⊂YA,B⊂Y be continuous images of the space X   and let f:X→A be a fixed continuous surjection. When is the inequalitydH(A,B)⩽inf{dsup(f,g):g∈C(X,Y),g(X)=B} replaced by the equality? The main result (Theorem 4.1) states that if X is a metric space of type (S) (see Definition 2.1) and A and B are its continuous images, then the equality holds for a completely arbitrarily fixed surjection f.