Analysis seminar: Gerald Beer (CSULA)

Title: A crash course in Bornologies

Abstract: By a bornology on a nonempty set X, we mean a family of subsets that contains the singletons, that is stable under finite unions, and that is stable under taking subsets. The prototype for a bornology is the so-called metric bornology: the family of metrically bounded subsets of a metric space. Bornologies help us to understand large structure. We enumerate some basic bornologies and give a few applications. We give an old result of S.-T. Hu characterizing the bornologies on a metrizable space that are metric bornologies with respect to some compatible metric, and give a fairly recent result of J. Cabello-Sanchez characterizing those metric spaces (X,d) for which UC(X,R) is a ring. We introduce the notion of bornological convergence of a sequence or net of closed subsets, of which Attouch-Wets convergence is the prototype, and give two applications to functional analysis.