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Kelley doctoral dissertation 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 hyper-space 2, and its hyperspaces. The reader is referred to for an outline of history and for a further information in this area. 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 Hausdorfi metric. An example of this is the study of hyperspaces. Hyperspaces: Fundamentals and Recent Advances (Chapman & Hall/CRC Pure and Applied Mathematics) by Alejandro Illanes () Alejandro Illanes Sam Nadler ISBN: Kostenloser Versand für alle Bücher mit Versand und Verkauf duch Amazon. Topologies on these and other families of subsets of a topological space X were studied by E. Often for understanding a structure, other closely related structures with the former are associated. In case when X is a metric space, the family of all bounded nonempty closed subsets of X can be metrized by the Hausdorfi metric (distance), in-troduced by Hausdorfi in 1914. that compactness (similarly connectedness) of 2, is equivalent to that of X. Topics include the topology for hyperspaces, examples of geometric models for hyperspaces, 2x and C(X) for Peano continua X, arcs in hyperspaces, the shape and contractability of hyperspaces. Vietoris proved the most basic facts of the structure of 2, as e. Presents hyperspace fundamentals, offering a basic overview and a foundation for further study. Gi-ven a topological space X, the hyperspace 2, of all nonempty closed subsets of X is equipped with the Vietoris topology, also called the exponential topo-logy, see or the finite topology, see, introduced in 1922 by Vietoris. Nadler Jr., Cones that are cells, and an application to hyperspaces 98 (1999. Moreover, we define a natural boundary function of hyperspaces, and we characterize the simple closed curve as the unique arcwise connected continuum for. Porti, The boundary of the Gieseking tree in hyperbolic three-space 93 (1999) 219259 Ancel, F.D. Hyperspace theory has its beginnings in the early years of XX century with the work of Felix Hausdorfi (1868-1942) and Leopold Vietoris (1891-2002). Calbrix, On the coincidence of the upper Kuratowski topology with the cocompact topology 93 (1999) 207218 Alperin, R.C., W.