The Category of H-modules over a Spectrum 15

f (x,t) = f (x,l) = g(x,0) = g(x,t) for all t e I, f e W(I,U) and

g e W(I,U). It is obvious that

p.(W(I,U) x W(I,U) x (0,1] x (0,1]) cW(K,U). Thus p„ is continuous

4 c c c 4

at (f,g,l,l).**

A pointed space is a pair (X,x ), where X is a space and x is a

point of X called the base point. We will often denote a pointed space

(X,x ) by X, and we will occasionally use * to denote the base point of

one or more spaces. We assume unless otherwise stated that a map f: X -» • Y

of pointed spaces is such that f(x0) = y . A subspace A of a pointed

space X is assumed to have the same base point as X. If A is a sub-

space such that X/A is Hausdorff, then X/A is also a space [6; 2.6] with

base point [A]. For pointed spaces X and Y, denote C((X,x ),(Y,y ) )

by C (X,Y) . Denote k(C (X,Y)) by Map (X,Y). If A and B are

subspaces of pointed spaces X and Y, respectively, let C_( (X,A),(Y,B))

be the subspace of C((X,A), (Y,B)) consisting of those maps f such that

f(xQ) = y. Denote k (CQ((X,A),(Y,B))) by MapQ((X,A),(Y,B)).

1.6. Proposition. If X, Y, and Z are pointed spaces and f: Y • Z is

a map, then f.: Map^(X,Y) -*Map^(X,Z) and f*: Map„(Z,X) +Map_(Y,X) are

x

u u u u

continuous.

Proof. This follows from (1.1) and the fact that Map (X,Y) is a

subspace of Map(X,Y).**

1.7. Proposition. Let X and Y be pointed spaces with subspaces A

and B, respectively. Define p : Map ( (X,A) , (Y,B) ) - Map (A,B) by

p,_(f) = f | : A • B. Then p,. is continuous.