Openness and monotoneity of induced mappings

Abstract

Abstract. It is shown that for locally connected continuum X if the induced mapping C(f ) : C(X) → C(Y ) is open, then f is monotone. As a corollary it follows that if the continuum X is hereditarily locally connected and C(f ) is open, then f is a homeomorphism. An example is given to show that local connectedness is essential in the result.All spaces considered in this paper are assumed to be metric. A mapping means a continuous function. We denote by N the set of all positive integers, and by C the complex plane. Given a space S, a point c ∈ S and a number ε > 0, we denote by B S (c, ε) the open ball in S with center c and radius ε.A continuum means a compact connected space. Given a continuum X with a metric d, we let 2 X denote the hyperspace of all nonempty closed subsets of X equipped with the Hausdorff metric H defined by H(A, B) = max{sup{d(a, B) : a ∈ A}, sup{d(b, A) : b ∈ B}}