![]() Some of the results of the study of basic properties such as compactness, connectedness, and separation axioms are that X is regular if and only if (iff) 2 X is Hausdorff X is completely regular iff 2 X is a Stone space X is normal iff 2 X is completely regular X is compact iff 2 X is compact X is compact metrizable iff 2 X is compact metrizable. The main themes in studying hyperspaces of a space X are the investigation of the properties of X that are carried over to 2 X, K(X) or F(X), and the determination whether 2 X, K(X) or F(X) belong to C or not if X belongs to C for a class C of generalized metric spaces or spaces with some special properties. /rebates/2fHyperspaces-Fundamentals-and-Recent-Advances2fIllanes-Nadler2fp2fbook2f9780824719821&. ![]() In addition, a.Abstract: The hyperspace consists of all the su. Based on this norm, the concept of Cauchy sequence can be similarly defined. The first step toward topologizing 2 X is defining a metric Ρ H on 2 X in the case when X is a bounded metric space this is called the Hausdorff metric. This normed hyperspace is clearly not a conventional normed space. When these collections are topologized, they are called hyperspaces of X. The first step toward topologizing 2X is defining a metric H on 2X in. ![]() To define the various topologies on the collection of closed subsets of a space X, certain collection of sets are considered. To define the various topologies on the collection of closed subsets of a space X, certain collection of sets are considered. All spaces are assumed to be Hausdorff topological spaces. All spaces are assumed to be Hausdorff topological spaces.
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