Connectedness and the hyperspace of metric spaces.

Abstract

One of the prime motivations for studying hyperspaces of a metric space is to understand the original space itself. The hyperspace of a metric space X is the space 2X of all non-empty closed bounded subsets of it, endowed with the Hausdorff metric. Our purpose is to study, in particular, connectedness properties of X and its hyperspace. We shall be concerned with knowing if a property P is extensional, that is, if X has property P then so does the hyperspace, or if a property is P is re ective, that is, if the hyperspace has property P then so does X itself. The hyperspace 2X and its subspace C(X) will be the focus of our study. First the Hau- dorff metric, p, is considered and introduced for the hyperspace 2X which is also inherited by C(X). As in (Nadler; [8]), when X is a continuum, the property of compactness is shown to be extensional to 2X and C(X). This is further generalised, when it is shown that each of 2X and C(X) is arcwise connected and hence are each arcwise connected continua, when X is a continuum. The classical results, the Boundary Bumping Theorems (due to Janiszewski [4]), which provide the required conditions under which the component of a set intersects its boundary, is proved using the Cut Wire Theorem (Whyburn; [13]). As an ap- plication, the Boundary Bumping Theorem (for open sets) is used to show the existence of continua arising out of convergence, in the Continuum of Convergence Theorem(Nadler; [8]). Using a construction of Whitney( [12]), the existence of a Whitney map, , for 2X and ! for C(X) are given. Using u, a special function o : [0; 1] -! 2X (due to Kelley [3]) called a segment is considered in the study of the arc structure of 2X and C(X). The equivalence of the existence of an order arc in 2X and the existence of a segment in 2X is also shown. A segment homotopy is then utilised to show that if one of 2X or C(X) is contractible then so is the other. This is presented in the Fundamental Theorem of Contractible Hyperspaces. The relationship between local connectedness and connectedness im kleinen is examined in order to understand the properties of Peano continua. Property S, introduced by Sierpin- ski( [10]), is considered and its connection to local connectedness is examined. Furthermore, a result of Wojdyslawski( [15]), which shows that local connectedness is an extensional prop- erty of a continuum X to the hyperspaces 2X and C(X), is given. Local connectedness is also re ective if either 2X or C(X) is a locally connected metric continuum. Lastly, Property K, by Kelley( [3]) is examined and is shown to be a sufficient condition for a continuum X to have its hyperspaces 2X and C(X) to be contractible. Consequently, if X is a Peano continuum then 2X and C(X) are contractible.