Compact-open Topology - Definition

Definition

Let X and Y be two topological spaces, and let C(X,Y) denote the set of all continuous maps between X and Y. Given a compact subset K of X and an open subset U of Y, let V(K,U) denote the set of all functions ƒ ∈ C(X,Y) such that ƒ(K) ⊂ U. Then the collection of all such V(K,U) is a subbase for the compact-open topology on C(X,Y). (This collection does not always form a base for a topology on C(X,Y).)

When working in the category of compactly-generated spaces, it is common to modify this definition by restricting to the subbase formed from those K which are the image of a compact Hausdorff space. Of course, if X is compactly generated and Hausdorff, this definition coincides with the previous one. However, the modified definition is crucial if one wants the convenient category of compactly-generated weak Hausdorff spaces to be Cartesian closed, among other useful properties. The confusion between this definition and the one above is caused by differing usage of the word compact.

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