10 Jack Palmer Sanders

projections. Hence p is continuous.**

Let I = [0,1] with the standard topology.

1.4. Proposition. Let X and Y be spaces. Let A be the subspace

of Map(xxi,Y) x Map(X*I,Y) x i consisting of those (f,g,a) such that

f(x,l) = g(x,0) for all x e X; if a = 0, then f(x,t) = f (x,l) for all

x e X, t e I; and if a = 1, then g(x,t) = g(x,0) for all x e X, t e I.

Define p : A -» - Map(X*I,Y) by

P (f,g,a)(x,t) = f(x,t/a), for 0 _ t £ a, a ^ 0;

= g(x,(t-a)/(l-a) ) , for a _ t _ 1, a^l .

Then p is continuous.

Proof. Let A c c(XI,Y) x c(Xxi,Y) x i be the same set A with

r c c

the relative topology. It is sufficient to show that the 'same function

p : A -

C(XXI,Y)

is continuous at each point (f,g,a). Suppose that

0 a 1, and let W(K,U) be a subbasic open neighborhood of p (f,g,a).

Define a map h : X x i + x x i by

h(x,t) = (x,t/a), for 0 _ t _ _ a;

= (x,l) , for a _ t _ 1.

For (x,t) e Kn(Xx[0,a]), f(h(x,t)) e U. There are an open set

A(x,t) c x, x e A(x,t), and d(x,t) 0 such that

[A(x,t)x(t-2d(x,t),t+2d(x,t))]n (Xxi) c

h""1(f_1

(U) ) . Denote

A(x,t) x (t-d(x,t),t+d(x,t)) by B(x,t). Since Kn (Xx[0,a]) is compact,

there is a finite set {(x.,t.),•••,(x ,t )} such that

1 1 n n

K n (Xx[0,a]) c B(x_,t_) U---U B(x ,t ). Let

1 1 n n