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 ) and ( Y , d ) be metric spaces, let A , B ⊂ Y be continuous images of the space X and let f : X → A be a fixed continuous surjection. When is the inequality d H ( A , B ) ⩽ inf { d sup ( 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.