Mrowka, Some comments on the space of subsets, Lecture Notes in Math., vol. However, local compactness is equivalent to the Fell topology being Hausdorff. condition of compactness to local compactness by studying topologies on the group () which make it a topological group and, at the same time, provide continuity of the evaluation function, that is, the so-called admissible group topologies. The Fell topology on the nonempty closed sets of a Hausdorff space is always compact, without any further assumption. That is, every point is interior to a compact set. Michael, Topologies on spaces of subsets, Trans. Local compactness means that every point has a compact neighborhood. The following statements are equiv-alent: (1) Every -compact set of X has a compact clo-sure in X (X is. A space X is compact if and only if CL(X) is compact With respect to countable compactness, in 1985 D. Hypercompactness is studied in relation to compactness and collective compactness of sets of mappings. Properties related to compactness in hyperspaces In 1951, E.A. Lin, Tychonoff’s theorem for the space of multifunctions, Amer. We give characterizations for a subspace of a hyperspace, endowed with either the Vietoris or the Wijsman topology, to be compact or relatively compact. Here a study is made of hyperspaces of Hausdorff completions and Hausdorff completions of hyperspaces, and, in particular, of a case where these spaces derive from two related uniform structures on a set. James Keesling, On the equivalence of normality and compactness in hyperspaces, Pacific J. We study the relations between a generalization of pseudocompactness, named (, M)-pseudocompactness, the countably compactness of subspaces of and the pseudocompactness of their hyperspaces. Frolik, Concerning topological convergence of sets, Czechoslovak Math. Fell, A Hausdorff topology for the closed subsets of a locally compact non-Hausdorff space, Proc. Chimenti, On the sequential compactness of the space of subsets, Bull.

Bushaw, Elements of general topology, Wiley, New York, 1963, Theorem 14, p. Birkhoff, Moore-Smith convergence in general topology, Ann. Conf., SUNY, Buffalo, N.Y., 1969) Lecture Notes in Mathematics, Vol. Mrówka, Some comments on the space of subsets, Set-Valued Mappings, Selections and Topological Properties of $2^X$ (Proc. Ernest Michael, Topologies on spaces of subsets, Trans.In this paper we present a short proof using known results concerning Hausdorff uniformities.

The known proofs use cardinality arguments and are long. Lin, Tychonoff’s theorem for the space of multifunctions, Amer. One of the most celebrated results in the theory of hyperspaces says that if the Vietoris topology on the family of all nonempty closed subsets of a given space is normal, then the space is compact (Ivanova-Keesling-Velichko). In another section of the paper, we have studied the local compactness of hyperspaces, endowed with either the Vietoris, the Wijsman, or the Hausdorff.

James Keesling, On the equivalence of normality and compactness in hyperspaces, Pacific J.

Hausdorff, Mengenlehre, 3rd ed., de Gruyter, Berlin, 1937 English transl., Chelsea, New York, 1962, p.

Zdeněk Frolík, Concerning topological convergence of sets, Czechoslovak Math. The present paper contains results characterizing relatively compact subsets of the space of the closed subsets of a metrizable space, equipped with various.

CrossRef View Record in Scopus Google Scholar. 20 (1972), 221–226 (English, with Russian summary). On the equivalence of normality and compactness in hyperspaces. Bushaw, Elements of general topology, John Wiley & Sons, Inc., New York-London, 1963.

Garrett Birkhoff, Moore-Smith convergence in general topology, Ann.We provethat f has shadowing (h-shadowing) if and only if 2f has shadowing(h-shadowing). We study properties related to first countability and countable compactness of uniformizable -hit-and-miss hyperspace topologies. We provethat f has shadowing (h-shadowing) if and only if 2f has shadowing(h-shadowing).ĪB - Given a nonempty compact metric space X and a continuousfunction f : X → X, we study shadowing and h-shadowing forthe induced maps on hyperspaces, particularly in symmetric products,Fn(X), and the hyperspace of compact subsets of X, 2X. N2 - Given a nonempty compact metric space X and a continuousfunction f : X → X, we study shadowing and h-shadowing forthe induced maps on hyperspaces, particularly in symmetric products,Fn(X), and the hyperspace of compact subsets of X, 2X. T1 - Shadowing for induced maps of hyperspaces

Compactness of hyperspaces