The Category of H-modules over a Spectrum 17

fAg(xAw) = f(x)Ag(W).

1.9. Proposition. For pointed spaces X, Y, and Z, the function

p : Map (Y,Z) - Map (YAX,ZAX) defined by p (f) = fAl is continuous.

Proof. Let r: Map (Y,Z) -*Map (Y,Z) xMap (X,X) be defined by

r(f) = (f,lX). Let TT: (ZxX,zUxXuZxxU) - (ZAX,*) be the projection. Let

q: Map ( (YxX,y xXuYxx ) , (ZAX,*) ) -*Map (YAX,ZAX) be the natural equivalence

of [6; 5.11], Then p = q ° u ^ O p o r and is therefore continuous.**

Given a space X, let X be the pointed space formed by adjoining a

disjoint base point * to X. For a pointed space X and h: I - X, let

h : I + X be defined by h (t) = h(t) for t e I, and h (*) = x . The

following result is clear.

1.10. Proposition. For a pointed space X, the correspondence

h - h yields an equivalence Map(I,X) - Map (I ,X).**

1.11. Proposition. Let X and Y be spaces. Let

A c Map (XAI ,y) x Map (XAi ,Y) x i be the subspace consisting of those

(f,g,a) such that f(xAl) = g (xAO) ; if a = 1, then g (xAt) = g (XAO) for

all t e I; if a = 0, then f(xAt) = f(xAl) for all t e I. Define

p8: A - Map0(XAi+,Y) by

p (f,g,a)(xAt) = f(xAt/a), for 0 t a, a ^ 0;

o — —

= g(xA(t-a)/(l-a)) , for a t 1, a ^ 1 •

Then p is continuous,

o

Proof. By applying the natural equivalence of [6; 5.11], one can