An Interesting Application of Classical Topology in Real Analysis

Dusan Repovs, University of Ljubljana


Using one of E.Michael's classical theorems on existence of continuous selections for multivalued mappings, we shall prove the following result from real analysis: Let (X, d) and (Y,h) be metric spaces and suppose that X is locally compact. Let C(X, Y) be the space of all continuous maps from X to Y, endowed with the usual topology of uniform convergence. Then there exists a continuous singlevalued function D from C(X, Y) x X x (0, \infty) to (0, \infty) such that for every (f,x,e) in C(X, Y) x X x (0, \infty) and for every x' in X we have d(x, x') < D(f,x,e) implies that h(f(x), f(x')) < e. Alternatively, this result can be proved using a classical theorem of C.H.Dowker on continuous separation of a lower and upper semicontinuous functions. As a corollary, we immediately obtain an elementary proof that the Cantor theorem on uniform continuity implies the Weierstrass theorem on boundedness of continuous functions on compacta. This was a joint work with P.V.Semenov. We shall also discuss subsequent joint work with J.Malesic on the modulus of continuity.