The Category of H-modules over a Spectrum 19

TT: YV(XAI ) - Cy(f) be theidentification map. Then g (TT (K) ) c u. Suppose

that K n TT(XAI) = $, theempty set, where A denotes theclosure of A.

Then TT~ (K) is compact in Y v (XAI ), g £ W(TT~ (K) ,U) , and

poq" (W(TT~ (K),U)) CW(K,U).

Suppose nowthat K n TT(XAI) ^ $. Since F iscompact, theevaluation

map e: F x [YV(XAI ) ] - Z is continuous by [6; 5.2]. F x Cy(f) is the

identification space obtained from F x (Yv(XAl )) by identifying (k,(xAl))

with (k,f(x)), by [6; 4.4], Since forany (k,(xAl)) we have

e(k,xAl) = e(k,f(x)), it follows that e induces a map e': FxCy(f) •* Z.

Since e'(g,Kmr (XAl) ) c u, there is foreach point x £ KriTT(XAl) a

neighborhood V of g anda neighborhood W of x such that

e' (V xw ) c u. Having covered K n TT(XAI) in this way, we obtain a finite

subcovering {w , • • • ,W }. Let V = V rv*»n V , and let

X.,

X

Xn

X

I n I n

W = W u-«-u W . Then e'(V*W) c U. Let K' = K-W. K' is compact, and

K' n TT(XAI) = $. it follows that V n W(TT (K'),U) is a neighborhood of g

in F. Forany k in this neighborhood, k(ir (W) ) c u since k e V, and

k(TT"" (K')) c U since k £ W(TT~ (K'),U). Since K c K' U W, it follows

that poq (k)£ W(K,U). Thus poq iscontinuous.**

1.13. Proposition. Let X, Y, Z, and W bepointed spaces, f: X - Y

a map. Let A be thesubspace of Map (ZAY,W) X Map (ZAXAI ,W) consisting

of those (g,h) such that g(zAf(x)) = h(zAxAl). Define

pg: A • MapQ(ZACy(f),W) by pg(g,h)(zAy) =g(zAy); pg(g,h) (zAxAt) =h(zAXAt)

Then p

q

iscontinuous.

Proof. There is the map lAf: ZAX- ZAY. Onecanreadily verify that

ZACy(f) = Cy(lAf). Theresult follows from (1.12).**