Recent Research in Hyperspace Theory

Hyperspace theory has its beginnings in the early years of XX century with the work of Felix Hausdorff (1868-1942) and Leopold Vietoris (1891-2002). Given a topological space X, the hyperspace 2X of all nonempty closed subsets of X is equipped with the Vietoris topology, also called the exponential topology, see [37, p. 160] or the finite topology, see [48, p. 153], introduced in 1922 by Vietoris [60]. Vietoris proved the most basic facts of the structure of 2X , as e.g. that compactness (similarly connectedness) of 2X is equivalent to that of X. In case when X is a metric space, the family of all bounded nonempty closed subsets of X can be metrized by the Hausdorff metric (distance), introduced by Hausdorff in 1914, [24]. Topologies on these and other families of subsets of a topological space X were studied by E. Michael in [48]. In particular, it is shown in that paper that if X is metric and compact, then the Vietoris topology coincides with the one introduced by the Hausdorff metric, [48, Proposition 3.5, p. 160]. The reader is referred to [10, Chapter 12, p. 750] for an outline of history and for a further information in this area. Since 1942, when J. L. Kelley doctoral dissertation [36] was published, the hyperspace theory became an important way of obtaining information on the structure of a topological space X by studying properties of the hyperspace 2X and its hyperspaces. The general task in this part of topology can be formulated as studying various properties of the hyperspaces to get more information about the structure and properties of the space itself. Since for a given space X the structure of the hyperspace 2X and its subspaces is rather complicate and hard to be seen, in particular any geometrical models