8 Jack Palmer Sanders
1. Preliminaries.
A topological space X is compactly generated if X is Hausdorff and
if each subset that intersects every compact set in a closed set is itself a
closed set. Such spaces are also called k-spaces. The category of compactly
generated spaces and continuous functions has been investigated by Steenrod
[6], among others. We adopt much of Steenrod's notation and assume all of
his results. Let CG denote this category.
Let X be a Hausdorff space. Define a space k(X) e CG as follows:
k(X) has the same underlying set, and M c k(X) is closed if M n C is
closed for all compact C c x. Given X and Y in CG, let
X x y = k(X*nY), where x denotes the product with the usual Cartesian
topology. If A is a subset of X and X e CG, we define the subspace A
of X to be k(A ), where A denotes the set A with the relative
r r
topology.
For Hausdorff spaces X and Y, let C(X,Y) denote the space of
continuous functions X - Y with the compact-open topology. For K compact
in X and U open in Y, let W(K,U) denote the set
{f e C(X,Y)|f(K) c u}, one of the subbasic open sets of C(X,Y). Denote
k(C(X,Y)) by Map(X,Y). If A c x and Bey , then C ( (X,A) , (Y,B))
denotes the subspace of C(X,Y) consisting of those continuous functions f
such that f(A) c B. Denote k(C((X,A), (Y,B))) by Map((X,A), (Y,B)). If
f: Y Z is a continuous function, then f : Map(X,Y) - Map(X,Z) is
defined by f*(g) = f ° g, and f*: Map(Z,X) - Map(Y,X) is defined by
f*(g) = g o f. Henceforth the terms "space" and "map" will be used to mean
a compactly generated space and a continuous function, respectively. We
need the following result [6; 5.9],
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