On a problem related to a non-squeezing theorem

In this paper we give a solution to a problem of Kulpa about the interior of the image of certain continuous maps f:X→Rn where X is a compact subset of Rn with non-empty interior. Moreover we show that the image of every continuous map f:X→R2 where X is a non-empty compact subset of R2 and diamf−1f(x)<3aX for every x∈FrX, has non-empty interior.