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 invented by Ralph Fox in 1945[1].

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).)

Properties

• If * is a one-point space then one can identify C(*,X) with X, and under this identification the compact-open topology agrees with the topology on X

• If Y is T₀, T₁, Hausdorff, regular, or Tychonoff, 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 {ƒ_n} converges to ƒ in the compact-open topology if and only if for every compact subset K of X, {ƒ_n} converges uniformly to ƒ 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, with Y locally compact Hausdorff (or even just preregular), then the composition map C(Y,Z) × C(X,Y) → C(X,Z), given by (ƒ,g) ↦ ƒ ○ g, is continuous (here all the function spaces are given the compact-open topology and C(Y,Z) × C(X,Y) is given the product topology).

• If Y is a locally compact Hausdorff (or preregular) space, then the evaluation map e : C(Y,Z) × Y → Z, defined by e(ƒ,x) = ƒ(x), is continuous. This can be seen as a special case of the above where X is a one-point space.

• 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(ƒ,g) = sup{d(ƒ(x), g(x)) : x in X}, for ƒ, g in C(X,Y).

Fréchet differentiable functions

Let X and Y be two Banach spaces defined on the same field, and let $\mathcal{C}^m\left(U,Y\right)$ denote the set of all m-continuously Fréchet-differentiable functions from the open subset $U\subseteq X$ to Y. The compact-open topology is the initial topology induced by the seminorms

$p_{K}\left( f \right) = \sup \{ \| D^{j}f\left( x \right)\|, x\in K, 0\leq j \leq m \}$

where $D^{0}f\left( x \right) = f\left( x \right)$, for each compact subset $K\subseteq U$.

See also

• Bounded-open topology

References

• Dugundji, J. (1966), Topology, Allyn and Becon, ISBN B000-KWE22-K 

• O.Ya. Viro, O.A. Ivanov, V.M. Kharlamov and N.Yu. Netsvetaev (2007) Textbook in Problems on Elementary Topology.
• compact-open topology on PlanetMath