Exponential Law for Uniformly Continuous Proper Maps

EXPONENTIAL LAW FOR UNIFORMLY CONTINUOUS PROPER MAPS R . AYALA, E . DOMINGUEZ, A . QUINTERO 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 continuous 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 C p (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 represent 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(-, -) . This work has been supported in part by CAICYT grant 0812-84 124 R. AYALA, E. DOMINGUEZ, A . QUINTERO Theorem . Let X, Y, Z metric spaces . We can define an injective map ID : Cup (X x Y, Z) -) Cup (X, Cüp (Y, Z)) as 1(f) (x) (y) = f (x, y) . If X is compact, then 1 is onto . Proof: It is easy to see that : (a) For each x E X the map oD(f)(x) is a up-map, because it is the composition of two up-maps . (b) oD (f) is a u-map, because f is a u-map . (c) Let us see that 4>(f) is a p-map. Given a compact subspace K C Cúp (Y,Z), we only have to prove that any sequence {xn} in 1(f)-1 (K) has a cluster point . Let {z,,} be a subsequence of {xj such that {-D(f)(zn )} is convergent ; let B E K be the limit point . Then, for each yo E Y the sequence {f (zn, yo)} converges to B(yo) . Consequently, H = {f(zn, yo) ; n E N}U{e(yo)} is a compact subspace of Z . This implies that the sequence {x,,} has a cluster point . Hence -P is well defined and it is an injective map . Since X is compact, each continuous map defined on X is also uniformly continuous . Then, given a continuous map