18 Jack Palmer Sanders

identify Map (XAI ,Y) with Map((Xxi,x xi),(y,y )), a subspace of

Map(Xxi,Y). The result follows from (1.4).**

Let X and Y be pointed spaces. Denote by X v Y the pointed space

obtained from X u Y by identifying x and y . Given a map f: X • Y,

denote by Cy(f) the pointed space obtained from Y v (XAI ) by identifying

x A 1 with f(x) Y is a closed subspace of Cy(f), and the inclusion

Y c Cy(f) is a homotopy equivalence. The map q: X - Cy(f) defined by

q(x) = XAO identifies X with a closed subspace X A 0 of Cy(f).

Cy(f)/XAO is Hausdorff, and hence C(f) = Cy(f)/XAO is a space. Y is a

closed subspace of C(f). Cy(f) and C(f) are the mapping cylinder and

mapping cone of f, respectively.

1.12. Proposition. Let X, Y, and Z be pointed spaces, f: X - Y a

map. Let A be the subspace of Map (Y,Z) x Map (XAI , Z) consisting of

those (g,h) such that g(f(x)) = h(xAl). Define p: A • Map (Cy(f) ,Z) by

p(g/h)(y) = g (y) ; p(g,h)(xAt) = h(xAt). Then p is continuous.

Proof. It suffices to show that p: A • C^(Cy(f),Z) is continuous on

r 0

compact sets by [6; 3.2], where A c C (Y,Z) x C (XAI ,Z) is A with the

relative topology. It is easily seen that the function

q: C (Y,Z) x^

CQ(XAI+,Z)

•*CQ

(YV(XAI+),Z)

defined by q(g,h)(y) = g(y),

q(g,h)(xAt) = h(xAt) is a homeomorphism. Using q we identify A with

the set B c C (YV(XAI ),Z) consisting of those k such that

k(xAl) = k(f(x)), where B has the relative topology. We show that

p°q : B - » C (Cy(f),Z) is continuous on compact sets.

Let F be a compact set of B, g e F, and let W(K,U) be a subbasic

open set in C (Cy(f),Z) which contains p(q (g)). Let