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.<ref>Template:Cite journal</ref>

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.<ref>Template:Cite book</ref>

Definition

Let Template:Mvar and Template:Mvar be two topological spaces, and let Template:Math 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 Template:Math denote the set of all functions Template:Math such that Template:Math In other words, <math>V(K, U) = C(K, U) \times_{C(K, Y)} C(X, Y)</math>. Then the collection of all such Template:Math is a subbase for the compact-open topology on Template:Math. (This collection does not always form a base for a topology on Template:Math.)

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.<ref>Template:Cite journal</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> 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 <math> X \times - </math> from the category of topological spaces always has a right adjoint <math> Hom(X, -) </math>. 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 Template:Math is a one-point space then one can identify Template:Math 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 Template:Math can be identified with the cartesian product of Template:Math copies of Template:Mvar and the compact-open topology agrees with the product topology.
• If Template:Mvar is Template:Math, Template:Math, 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 Template:Mathis a subbase for the compact-open topology on Template:Math.<ref>Template:Cite journal</ref>
• 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 Template:Mathconverges to Template:Math in the compact-open topology if and only if for every compact subset Template:Mvar of Template:Mvar, Template:Mathconverges uniformly to Template:Math 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 Template:Math and Template:Mvar are topological spaces, with Template:Mvar locally compact Hausdorff (or even just locally compact preregular), then the composition map Template:Math given by Template:Math is continuous (here all the function spaces are given the compact-open topology and Template:Math is given the product topology).
• If Template:Mvar is a locally compact Hausdorff (or preregular) space, then the evaluation map Template:Math, defined by Template:Math, 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 Template:Math is metrizable, and a metric for it is given by Template:Math for Template:Math in Template:Math. 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 can be used to topologize the following sets:<ref name=":0">Template:Cite book</ref>

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

In addition, there is a homotopy equivalence between the spaces <math>C(\Sigma X, Y) \cong C(X, \Omega Y)</math>.<ref name=":0" /> The topological spaces <math>C(X,Y)</math> 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 mapsTemplate:Clarify <math display=block>\pi(X,Y) = \{[f]: X \to Y \mid f \text{ is a homotopy class}\}.</math> This is because <math>\pi(X,Y)</math> is the set of path components in <math>C(X,Y)</math>Template:Endashthat is, there is an isomorphism of sets <math display=block>\pi(X,Y) \to C(I, C(X, Y))/{\sim},</math> where <math>\sim</math> is the homotopy equivalence.

Fréchet differentiable functions

Let Template:Mvar and Template:Mvar be two Banach spaces defined over the same field, and let Template:Math denote the set of all Template:Mvar-continuously Fréchet-differentiable functions from the open subset Template:Math to Template:Mvar. The compact-open topology is the initial topology induced by the seminorms

<math>p_{K}(f) = \sup \left\{ \left\| D^j f(x) \right\| \ : \ x \in K, 0 \leq j \leq m \right\}</math>

where Template:Math, for each compact subset Template:Math.Template:Clarification needed

See also

• Topology of uniform convergence
• Template:Annotated link

References

• Template:Cite book
• 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