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

Given a set-valued map T from a set X into a metric space Y, consider the ``\epsilon-enlargement'' of T, i.e. the set-valued map \epsilonT from X to Y which associates to each x in X the set of all points whose distance from T(x) is less than \epsilon. Moreover, call T ``coherent'' if for every x in X, there exists a neighbourhood W of x such that \cap _{w in W} T(w) =/= \emptyset. When X is paracompact, Y is a normed linear space and T takes convex values, Deutsch and Kenderov show that \epsilonT has a continuous selection for each \epsilon > 0 if and only if it is coherent for each \epsilon > 0. We study the relation between coherence and existence of continuous selections in general. We also investigate the existence of \epsilon-continuous selections, since for Y complete, limits, as \epsilon goes to zero, of sequences of \epsilon-continuous selections for \epsilonT are continuous selections for T.

Date received: June 3, 2003