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).)

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.

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