V. Gregoriades (Darmstadt): The complexity of the set of points of continuity of a multi-valued function
Donnerstag, 08.12.2011 14:15 im Raum N 2
We consider a notion of continuity concerning multi-valued functions between metric spaces. We focus on multi-valued functions assigning points to nonempty closed sets and we answer the question asked by M. Ziegler (Real Computation with Least Discrete Advice: A Complexity Theory of Nonuniform Computability with Applications to Effective Linear Algebra, to appear in Ann. Pure Appl. Logic) of determining the complexity of the set of points of continuity of such a multi-valued function. We will see that, under appropriate compactness assumptions, it is Π02. Under a natural weakening of these assumptions it can be a true Σ03 set, while in the general case it might not even be Borel. In particular, this notion of continuity is not metrizable, in the sense that if we view a multi-valued function F from X to Y as a single-valued function from X to the family F(Y) of closed subsets of Y, it is not necessarily the case that there is a metrizable topology on F(Y) such that F is continuous at an arbitrary x∈X in the usual sense (of single-valued functions) exactly when it is continuous at x in the sense of multi-valued functions. We will also present some similar results about a stronger notion of continuity, which is closely related to the Fell topology and thus in some cases turns out to be metrizable.