Compact-open topology

Template:Short description 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.

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.

Definition

Let Template:Mvar and Template:Mvar be two topological spaces, and let C(X, Y) denote the set of all continuous maps between Template:Mvar and Template:Mvar. Given a compact subset Template:Mvar of Template:Mvar and an open subset Template:Mvar of Template:Mvar, let V(K, U) denote the set of all functions  f  ∈ C(X, Y) such that  f (K) ⊆ U. In other words, $V(K,U)=C(K,U){\times }_{C(K,Y)}C(X,Y)$. 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 Template:Mvar that are the image of a compact Hausdorff space. Of course, if Template:Mvar 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.

If Template:Mvar is locally compact, then $X\times -$ from the category of topological spaces always has a right adjoint $Hom(X,-)$. This adjoint coincides with the compact-open topology and may be used to uniquely define it. The modification of the definition for compactly generated spaces may be viewed as taking the adjoint of the product in the category of compactly generated spaces instead of the category of topological spaces, which ensures that the right adjoint always exists.

Properties

• If * is a one-point space then one can identify C(*, Y) with Template:Mvar, and under this identification the compact-open topology agrees with the topology on Template:Mvar. More generally, if Template:Mvar is a discrete space, then C(X, Y) can be identified with the Cartesian product of |X| copies of Template:Mvar and the compact-open topology agrees with the product topology.
• If Template:Mvar is T₀, T₁, Hausdorff, regular, or Tychonoff, then the compact-open topology has the corresponding separation axiom.
• If Template:Mvar is Hausdorff and Template:Mvar is a subbase for Template:Mvar, then the collection {V(K, U) : U ∈ S, K compact} is a subbase for the compact-open topology on C(X, Y).
• If Template:Mvar is a metric space (or more generally, a uniform space), then the compact-open topology is equal to the topology of compact convergence. In other words, if Template:Mvar is a metric space, then a sequence { f_n } converges to  f  in the compact-open topology if and only if for every compact subset Template:Mvar of Template:Mvar, { f_n } converges uniformly to  f  on Template:Mvar. If Template:Mvar is compact and Template:Mvar is a uniform space, then the compact-open topology is equal to the topology of uniform convergence.
• If X, Y and Template:Mvar are topological spaces, with Template:Mvar locally compact Hausdorff (or even just locally compact preregular), then the composition map C(Y, Z) × C(X, Y) → C(X, Z), given by ( f , g) ↦  f ∘ 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 Template:Mvar is a locally compact Hausdorff (or preregular) space, then the evaluation map e : C(X, Y) × X → Y, defined by e( f , x) =  f (x), is continuous. This can be seen as a special case of the above where Template:Mvar is a one-point space.
• If Template:Mvar is compact, and Template:Mvar is a metric space with metric Template:Mvar, then the compact-open topology on C(X, Y) is metrizable, 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). More generally, if Template:Mvar is hemicompact, and Template:Mvar metric, the compact-open topology is metrizable by the construction linked here.

Applications

The compact open topology (or the k-ification of it) can be used to topologize the following sets:

• $\mathrm{\Omega }(X,x_0)=\{f:I\to X\mid f(0)=f(1)=x_0\}$, the loop space of $X$ at $x_0$,
• $E(X,x_0,x_1)=\{f:I\to X\mid f(0)=x_0\text{ and }f(1)=x_1\}$,
• $E(X,x_0)=\{f:I\to X\mid f(0)=x_0\}$.

In addition, there is a homotopy equivalence between the spaces $C(\mathrm{\Sigma }X,Y)\cong C(X,\mathrm{\Omega }Y)$. The topological spaces $C(X,Y)$ are useful in homotopy theory because they can be used to form a topological space and a model for the homotopy type of the set of homotopy classes of maps $$\pi (X,Y)=\{[f]:X\to Y\mid f\text{ is a homotopy class}\}.$$ This is because $\pi (X,Y)$ is the set of path components in $C(X,Y)$

1. REDIRECT Template:En dash

Template:R protectedthat is, there is an isomorphism of sets $$\pi (X,Y)\to C(I,C(X,Y))/\sim ,$$ where $\sim$ is the homotopy equivalence.

Fréchet differentiable functions

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

$p_K(f)=\sup \{\Vert D^{j}f(x)\Vert :x\in K,0\le j\le m\}$

where D⁰ f (x) =  f (x), for each compact subset K ⊆ U.Template:Clarification needed

See also

• Topology of uniform convergence
• Template:Annotated link

References

• O.Ya. Viro, O.A. Ivanov, V.M. Kharlamov and N.Yu. Netsvetaev (2007) Textbook in Problems on Elementary Topology.
• Template:Planetmath reference
• Topology and Groupoids Section 5.9 Ronald Brown, 2006