Compact-open topology

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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.

[edit] 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).)

[edit] Properties

• If Y is T₀, T₁, Hausdorff, or regular, then the compact-open topology has the corresponding separation axiom.

• If X is Hausdorff and S is a subbase for Y, then the collection {V(K, U) : U in S} is a subbase for the compact-open topology on C(X, Y).

• 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 compact convergence. In other words, if Y is a uniform space, then 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. In particular, if X is compact and Y is a uniform space, then the compact-open topology is equal to the topology of uniform convergence.

• If X, Y and Z are topological spaces, and if X is a locally compact regular space (not necessarily Hausdorff), then the composition map C(Z, X) × C(X, Y)  →  C(Z, Y), given by (f, g) gof, is continuous, where all the function spaces are given the compact-open topology, and where C(Z, X) × C(X, Y) is given the product topology. In particular, if X is a locally compact regular space, then the evaluation map e : X × C(X, Y) → Y defined by e(x, f) = f(x) is continuous, where X is regarded as the function space C(∗, X) of maps from the one-point space to X.

• If X is compact, and if Y is a metric space with metric d, then the compact-open topology on C(X, Y) is metrisable, and a metric for it is given by e(f, g) = sup{d(f(x), g(x)) : x in X}, for f, g in C(X, Y).