Homotopical properties of hyperspaces of generalized continua: the proper and ordinary cases

Given a generalized continuum X, let CLF(X) and C(X) denote its hyperspaces of (non-empty) closed subsets and subcontinua, respectively, with the Fell topology (= Vietoris topology on C(X)). This paper deals with the connectedness at infinity and the proper homotopy of these hyperspaces. It is proved that both CLF(X) and C(X) are one-ended generalized continua. Moreover, CLF(X) is always strongly one-ended, while for C(X) this property is equivalent to the continuumwise connectedness of X. Concerning homotopy types, we show that CLF(X) is always contractible. By contrast, its proper homotopy type is trivial if and only if X is cik at infinity. Furthermore, CLF(X) is properly homotopically trivial if and only if it has trivial weak proper homotopy type. With regard to C(X), the triviality of the proper homotopy type is shown to be equivalent to contractibility, and this is attained whenever the generalized continuum X has the Kelley property. In addition, C(X) has trivial weak homotopy type (proper or ordinary) if and only if X is continuumwise connected. Proper and ordinary hyperspace (deformation) retracts are also considered. For generalized Peano continua such retracts are characterized. In absence of local connectedness some results and examples are given and several questions are posed.