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 commonly studied in homotopy theory and functional analysis.

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 f in C(X, Y) such that f(K) is contained in U. Then the collection of all such V(K, U) is a subbase for the compact-open topology. (This collection does not always form a base for a topology on C(X, Y).)

Properties

• If Y is a uniform space (in particular, if Y is a metric space), then the compact-open topology is equal to the topology of uniform convergence on compact sets. In other words, in this case, a sequence {f_n} converges to f in the compact-open topology if and only if for every compact subset K of X, {f_n} converges uniformly to f on K. If, in addition, X itself is compact, then the compact-open topology is equal to the topology of uniform convergence.

• If X is a locally compact Hausdorff space, then the evaluation map e : X × C(X, Y)  →  Y defined by e(x, f) = f(x) is continuous.