16 Jack Palmer Sanders

Proof. Since Map ((X,A),(Y,B)) and Map (A,B) are subspaces of

Map((X,A),(Y,B)) and Map(A,B), respectively, it suffices to show that the

same function p : C((X,A),(Y,B)) - C(A,B ) is continuous at each point f,

where B c y is B with the relative topology. Let W(K,U) be a

subbasic open set in C(A,B ) with p^f) e W(K,U). U = BnV, where V is

open in Y. Hence W(K,V) is open in C(X,Y), and

f e W(K,V) n C((X,A) , (Y,B) ) , which is open in C((X,A),(Y,B)). Clearly

p (W(K,V)nC((X,A),(Y,B))) c W(K,U), and so p is continuous.**

1.8. Proposition. Let X and Y be pointed spaces. Let A and B

be subspaces of X and Y, respectively, such that X/A and Y/B are

Hausdorff. Define p : Map ((X,A),(Y,B)) - Map (X/A,Y/B) by

pr(f)[x] = [f(x)]. Then pr is continuous.

Proof. Let TT: (Y,B) - (Y/B,*) be the projection map. Let

q: Map ((X,A),(Y/B,*)) - Map (X/A,Y/B) be the natural equivalence of

[6; 5.11]. Then p is the composition

77*

MapQ((X,A), (Y,B)) • Map ((X,A),(Y/B,*)) -2_* Map (X/A,Y/B) ,

and hence p is continuous.**

Let X and Y be pointed spaces. The smash product X A Y is the

space obtained from X x Y by collapsing

xxYn u x

n

x Y to a P°intf

the base

point of X A Y. We will denote the image in X A Y of (x,y) by x A y.

The smash product in CG is commutative and associative. If B is a

closed subspace of Y such that Y/B is Hausdorff, then X A B is a

closed subspace of X A Y, and XA(Y/B) = XAY/XAB. If f: X - Y and

g: W - Z are maps, then there is a map fAg: XAW - YAZ defined by