Some Convergence Results for Partial Maps

Abstract

In this paper we analyze some aspects of a new notion of convergence for nets of partial maps, introduced in BCDL. In particular, we show that the introduced bornological convergence reduces to a natural uniform convergence relative to the bornology when the partial maps have a common domain. We then provide a new notion of upper convergence, which looks much more manageable than the original one. We show by examples that the two notions, though different in generalcases, do  agree for sequences of strongly uniformly continuous (relative tothebornology) partial maps. More generally, coincidence for nets and not only sequences is shown in case the target space of the maps is totally bounded. This last result is interesting in view of possible applications, since partial maps are usually utilityf unctions, thus when dealing with general models, monotone transformations valued in [0,1] give raise to the same utility unctions.