## An Interesting Application of Classical Topology in Real Analysis

### Dusan Repovs, University of Ljubljana

Abstract

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.