CONTRACTIBILITY OF THE HYPERSPACE OF SUBCONTINUA

Let X be a metric continuum. Denote by 2 X and C(X) the hyperspaces of nonempty closed subsets and nonempty subcontinua of X respectively and endow each with the Hausdorff metric D. In the present paper we investigate various questions concerning the contractibility of C(X). In particular, we resolve questions raised in [6] of characterizing the contractibility of C(X) in terms of homotopy equivalences and of characterizing the contractibility of C(X X Y). Also, the contractibility of the hyperspaces of continuous images of continua is studied. In order to establish the noncontractibility of a hyperspace, we prove that a simple condition about X is a sufficient condition for the noncontractibility of C(X). We remark that Wardle [9] has investigated maps which preserve property K (property K implies C(X) is contractible). Throughout the paper the symbol I will be reserved for the closed interval [0,1]. For a general as well as a particular reference on C(X), see [6]. 1. Preliminaries. We collect in this section some definitions and known facts. Finally, we prove a basic lemma. A map h: X X I- • 2 Y is called increasing if h(x,t) C h(x,t') for x E X and t < t'. A contraction f X in 2 X is a continuous homotopy h:X X I- • 2 X such that, for some A E 2 X, it is true that h(x,0) = {x) and h(x,1) = A for x • X. A contraction f X in C(X) is analogously defined where 2X is replaced by C(X). It is known [3] that a contraction f X in 2 X exists when and only when a contraction of X in C(X) exists. Moreover, the following theorems are known. THEOREM 1.1. [3, 3.1, page 25]. The following statements are equivalent. 1. A contraction f X in 2 X exists. 2. 2 X is contractible