Quasicontinuous functions, minimal usco maps and topology of pointwise convergence

In [HOLÁ, Ľ.—HOLÝ, D.: Pointwise convergence of quasicontinuous mappings and Baire spaces, Rocky Mountain J. Math.] a complete answer is given, for a Baire space X, to the question of when the pointwise limit of a sequence of real-valued quasicontinuous functions defined on X is quasicontinuous. In [HOLÁ, Ľ.—HOLÝ, D.: Minimal USCO maps, densely continuous forms and upper semicontinuous functions, Rocky Mountain J. Math. 39 (2009), 545–562], a characterization of minimal USCO maps by quasicontinuous and subcontinuous selections is proved. Continuing these results, we study closed and compact subsets of the space of quasicontinuous functions and minimal USCO maps equipped with the topology of pointwise convergence. We also study conditions under which the closure of the graph of a set-valued mapping which is the pointwise limit of a net of set-valued mappings, is a minimal USCO map.

A Note on Minimal Usco Maps

We prove that the composition of a minimal usco map, defined on a Baire space, with a lower semicontinuous function is single valued and usco at each point of a dense G$ subset of its domain. This extends earlier results of Kenderov and Fitzpatrick. As a first consequence, we prove that a Banach space, with the property that there exists a strictly convex, weak* lower semicontinuous function on its dual, is a weak Asplund space. As a second consequence, we present a short proof of the fact that a Banach space with separable dual is an Asplund space.