Openness of the induced map cn(f)

Abstract

Given a map between compact metric spaces f : X-- and gt;Y , it is possible to induce a map between the n-fold hyperspaces Cn(f) : Cn(X) -- and gt; Cn(Y ) for each positive integer n. Let A and B be classes of maps. A general problem is to find the interrelations between the following two statements:1. f 2 A; 2. Cn(f) 2 B. It is known that 1 and 2 are equivalentconditions if both A and B are the class of homeomorphisms. If A and B are the class of open maps, then the openness of Cn(f) implies the openness of f. Furthermore, there exists an open map f such that Cn(f) is not open. Moreover, if Cn(f) is open and n and gt; 3, then f is both open and monotone. Our main result is Theorem 3.2, where we prove that if the induced map Cn(f) is an open map, for n and gt;= 2, then f is a homeomorphism.