Quasicontinuous Functions, Densely Continuous Forms and Compactness

Abstract

ABSTRACT. Let X be a locally compact space. A subfamily F of the space D (X, R) of densely continuous forms with nonempty compact values from X to R equipped with the topology τ UC of uniform convergence on compact sets is compact if and only if sup(F ) : F ∈ F is compact in the space Q(X, R) of quasicontinuous functions from X to R equipped with the topology τ UC . IntroductionQuasicontinuous functions were introduced by K e m p i s t y in 1932 in [14]. They are important in many areas of mathematics. They found applications in the study of minimal USCO and minimal CUSCO maps [7], [8], in the study of topological groups [3], [16], [18], in proofs of some generalizations of Michael's selection theorem [5], in the study of extensions of densely defined continuous functions [6], in the study of dynamical systems [4]. The quasicontinuity is also used in the study of CHART groups [17].Densely continuous forms were introduced by H a m m e r and M c C o y in [12]. Densely continuous forms can be considered as set-valued mappings from a topological space X into a topological space Y which have a kind of minimality property found in the theory of minimal USCO mappings. In particular, every minimal USCO mapping from a Baire space into a metric space is a densely continuous form. There is also a connection between differentiability properties of convex functions and densely continuous forms as expressed via the subdifferentials of convex functions, which are a kind of convexification of minimal USCO mappings [12].