# Structure of hyperspaces

In this thesis we investigate contractibility and local contractibility in the hyperspace of subcontinua of continuum X. We show that C(X) is contractible if and only if it is freely contractible to the point X. Since free contractibility to a point in a continuum is equivalent to arc-smoothness at that point, this shows that C(X) is arc-smooth at the point X for any continuum X such that C(X) is contractible, answering a question of Goodykoontz We also show that if C(X) is contractible and (mu)('-1)(t(,0)) is irreducible between two points for some t(,0), then there is a map from C(X) onto (mu)('-1)(t(,0)). Furthermore, this map remains surjective when restricted to a separating subcontinuum of C(X). In certain cases, these theorems imply that C(X) cannot be embedded in E('3) We give conditions which imply the local contractibility of C(X). These conditions are very weak in the sense that it is not known whether or not there is a continuum for which they are not satisfied. It remains an open question whether or not C(X) is always locally contractible at the point X Duda has given a geometric construction for the hyperspace of subcontinua of a finite acyclic graph. We construct the hyperspace of a bouquet of circles by expressing it as a quotient space of the hyperspace of an acyclic graph. This construction is used to show that the property of being an ANR is not strongly Whitney reversible. Also, we are able to construct, as a quotient space of the hyperspace of an acyclic graph, the hyperspace of any finite graph. These results answer questions of Duda {3} and Nadler {13}