Contractibility of Whitney continua in C(X)

Abstract

We show that there are Whitney maps on the 2-cell such that Whitney continua in the hyperspace of the 2-cell are non-contractible, non-locally contractible, and have non-trivial Čhech cohomology in dimension 2. This implies that contractibility, local contractibility, being an AR, being an ANR, and acyclicity in Čech cohomology are not Whitney properties. We show, however, that contractibility is a Whitney property for the class of dendrites.