Compact-open topology

In mathematics, the compact-open topology is a topology defined on the set of continuous maps between two topological spaces. The compact-open topology is one of the commonly used topologies on function spaces, and is applied in homotopy theory and functional analysis. It was introduced by Ralph Fox in 1945.[1]

If the codomain of the functions under consideration has a uniform structure or a metric structure then the compact-open topology is the "topology of uniform convergence on compact sets." That is to say, a sequence of functions converges in the compact-open topology precisely when it converges uniformly on every compact subset of the domain.[2]

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 f  ∈ C(X, Y) such that f (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 that 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.[3][4][5] The confusion between this definition and the one above is caused by differing usage of the word compact.

The compact open topology can be used to topologize the following sets:[7]

Let X and Y be two Banach spaces defined over the same field, and let C m(U, Y) denote the set of all m-continuously Fréchet-differentiable functions from the open subset UX to Y. The compact-open topology is the initial topology induced by the seminorms