Abstract EXPONENTIAL LAW FOR UNIFORMLY CONTINUOUS PROPER MAPS

The purpose of this note is to prove the exponential law for uniformly continuous proper maps. Let X be a regular space and Y a locally compact regular space. It is well known that the spaces of continuousmaps C(X x Y, Z) and C(X, C(Y Z)) are homeomorphic considering the compact-open topology. This property has important consequences in the study of the path-components of the function spaces and in Homotopy Theory: The exponential law for the uniformly con-tinuous proper maps has similar consequences in some particular cases. All the spaces we consider, unless otherwise mentioned, are metric spaces. A proper map will. be a continuous map f: X-+ Y such that for every compact subspace K of Y, f-1(K) is á compact in X. To abbreviate,- we will say that f is a p-map. A u-map is a uniformly,cgntinuous map, and a up-map will be a uniformly continuous p-map. A up-isomorphism f l a homeomorphism such that f and f-1 are u-maps., By C(X, Y), Cp (X, Y) and Cp (X, Y) we will denote the sets of continuous maps, p-maps and up-maps between X and Y, respectively. With Cúp (X, Y) we will represent the space of up-maps with the topology of uniform convergente. In this note we prove that the up-maps follow the exponential law if X is compact; that is, the functors X x (-) and Cúp (X,-) are adjoint. We also prove that if X is not compact these functors are not generally adjoint. A up-homotopy (p-homotopy) between up-maps (p-maps is a homotopy which is a up-map (p-map. With [-,-], [-,-]P, and [-,-] p we will rep-resent the sets of homotopy, p-homotopy and up-homotopy respectively. Also, the corresponding homotopy classes will be denoted by [f], [f]p and [f]up. R" will stand for the n-dimensional Euclidean space, and I for the unit interval [0,1] with that distante. The Euclidean norm will be represented by- ~, and the distante of a metric space by d(-,-)