on the convergence of closed and compact sets - Project Euclid
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on the convergence of closed and compact sets - Project Euclid
9(X), 2X and Q(X) of all closed subsets, all non-empty closed subsets ... convergence, as defined by Choquet [2], which however is equivalent to the topological ...
PACIFIC JOURNAL OF MATHEMATICS Vol. 108, No. 1, 1983
ON THE CONVERGENCE OF CLOSED AND COMPACT SETS EVA LOWEN-COLEBUNDERS For a topological Hausdorff space X we study the hyperspaces 9(X), 2X and Q(X) of all closed subsets, all non-empty closed subsets and all non-empty compact subsets endowed with the convergence of sets. In this paper we shall work with the filter description of this convergence, as defined by Choquet [2], which however is equivalent to the topological convergence of nets of sets as defined by Frolik and Mrowka. We shall study the relation between properties of X and properties of the spaces 9(X), 2X and &(X) such as compactness, local compactness, regularity and the topological and pretopological character.
1. Introduction. The aim of this paper is to study properties of the convergence of closed or compact sets of a Hausdorff topological space. The class ^P( X) of all closed subsets of a Hausdorff topological space X, will be endowed with the natural pseudotopological structure of closed convergence. On 2X, the collection of non-empty closed subsets of X and on &(X)9 the collection of non-empty compact subsets of X, we consider the induced pseudotopological structures. The main purpose is to investigate under what conditions on X, the spaces 9(X)9 2X and G(X) are compact, locally compact, regular, topological or pretopological. The pseudotopological structure of closed convergence has been defined by Choquet in [2, p. 87] using the supremum (Sup) and the infimum (inf) of filters on (X). If x is a filter on
