Hyperspaces of a continuum

Introduction. Among the topological invariants of a space X certain spaces have frequently been found valuable. The space of all continuous functions on X and the space of mappings of X into a circle are noteworthy examples. It is the purpose of this paper to study two particular invariant spaces associated with a compact metric continuum X; namely, 2X, which consists of all closed nonvacuous subsets of X, and C(X), which consists of closed connected nonvacuous subsets('). The aim of this study is twofold. First, we wish to investigate at length the topological properties of the hyperspaces, and, second, to make use of their structure to prove several general theorems. If X is a compact metric continuum it is known that: 2X is Peanian if X is Peanian [7 ], and conversely [8]; 2x is always arcwise connected [1 ]; 2X is the continuous image of the Cantor star [4]; if X is Peanian, each of 2x and C(X) is contractible in itself [9]; and if X is Peanian, 2x and C(X) are absolute retracts [10]. In ??1-5 of this paper further topological properties are obtained. In particular: 2x has vanishing homology groups of dimension greater than 0, both hyperspaces have very strong higher local connectivity and connectivity properties-including local p-connectedness in the sense of Lefschetz for p > 0, and, the question of dimension is resolved except for the dimension of C(X) when X is non-Peanian. All of the results of the preceding paragraph for 2x are shown simultaneously for 2x and (X) in the course of the development. In ?6 a characterization of local separating points in terms of C(X) is obtained and a theorem of G. T. Whyburn deduced. In ?7 it is shown that for a continuous transformation f(X) = Y we may under certain conditions find XoCX, with XO closed and of dimension 0, such that f(Xo) = Y. In ?8 this result is utilized in the study of Knaster continua. In order that X be a Knaster continuum it is necessary and sufficient that C(X) contain a unique arc between every pair of elements. If there exist Knaster continua of dimension greater than 1 then there exist infinite-dimensional Knaster continua.