The Category of H-modules over a Spectrum

1.1. Proposition. If X, Y, and Z are spaces and f: Y - Z is a

map, then f^ and f* are continuous.**

1.2. Proposition. Let X, Y, and Z be spaces. The function

p : Map(X,Y) x z -Map(X,Yxz) defined by p (f,z)(x) = (f(x),z) is

continuous. The function q: Y x Map(X,Z) -

Map(X,Yxz)

defined by

q(y,f)(x) = (y,f(x)) is continuous.

Proof. For p , it suffices to show that the same function

p : C(X,Y) x z - C(X,Yx z) is continuous, by [6; 3.2, 4.5, and 5.3]. Ey

[4; XII, 5.1], {W(A,Ux V)|A compact in X, U open in Y, V open in Z} is a

subbasis for C(X,Yx z). Since (p )"

(W(A,UX

v)) = W(A,U)

x

v, which is

open in C(X,Y)

x

z, p is continuous. The proof for q is similar.**

Let f: X - Y and g: W -*Z be maps of spaces. By [6; 3.2], there

is a map f * g: X x w ^ Y x z defined by f x g(x,w) = (f(x),g(w)).

1.3. Proposition. Let X, Y, Z, and W be spaces with subspaces

A, B, C, and D, respectively. Define

p : Map((X,A),(Y,B)) x Map((W,D),(Z,C)) + Map((XxW,AxWuXxD),(YxZ,BxZuYxC))

by P9(f,g) = f

x

g. Then p is continuous.

Proof. By [6; 3.2, 4.5, and 5.3], it suffices to show that the same

function p : C ( (X,A) , (Y,B) ) x C ( (W,D) , (Z,C) ) -*

^2 c

C((Xxw,AXWUXXD),(Yx z,Bx ZuYx c)) is continuous. For this, it suffices to

c c c

show that p„: C(X,Y) x C(W,Z) - C(Xxw,Yx z) is continuous. As in (1.2),

2 c c

{w(K,Ux V)IK compact in X x w, U open in Y, V open in Z} is a subbasis

c

for C(Xxw,Yx Z). It is easily seen that

c

(pj" (W(K,Ux V)) = W(TT (K),U) x W(TT (K),V), where TT

T

and I T are the

2 c X c W X W