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
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