Formal n-convexity, n-paracompactness and continuous selections
by
Debora DiCaprio
York University (Canada) - Seconda Universita' degli Studi di Napoli (Italy)
Coauthors: Stephen Watson (York University)

Let X be a nonempty set and n be a positive integer or \infty. Denote by FCC_n(X) the set of all s in [0, 1]^X such that |supp(s)| < n+1 and \sum_{x in X} s(x) = 1; we call FCC_n(X), endowed with the induced uniform topology, ``the space of formal n-convex combinations over X". We define X to be ``formally n-convex" if a function p from FCC_n(X) into X with particular properties that will be discussed exists. A T₁-space X is called ``n-paracompact", if every open cover of X has a locally < n+1 partition of the unity subordinated to it. We show that if X is a n-paracompact space and (Y, d) is a complete metric n-convex space, then every l.s.c. set-valued map T from X to Y, whose values satisfy certain n-convexity properties, has a continuous selection. This selection theorem is stronger than the Michael's Selection Theorem, both the convex-valued version and the 0-dimentional one, always considered as two independent cases in the literature.

Date received: June 18, 2003

Copyright © 2003 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # calv-10.