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.
[1]
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.
[2]
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 ∈ C(X, Y) such that f (K) ⊆ U. In other words, . 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 that 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.
[3]
[4]
[5] The confusion between this definition and the one above is caused by differing usage of the word
compact.
If X is locally compact, then from the category of topological spaces always has a right adjoint . 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 Y, and under this identification the compact-open topology agrees with the topology on Y. More generally, if X is a
discrete space, then C(X, Y) can be identified with the
cartesian product of |X| copies of Y and the compact-open topology agrees with the
product topology.
- 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 ∈ S, K compact} is a
subbase for the compact-open topology on C(X, Y).
[6]
- If Y 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 Y is a metric space, then a sequence { fn }
converges to f in the compact-open topology if and only if for every compact subset K of X, { fn } converges uniformly to f on K. 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 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 X 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 X is a one-point space.
- If X is compact, and 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).
Applications
The compact open topology can be used to topologize the following sets:
[7]
- , the
loop space of at ,
- ,
- .
In addition, there is a
homotopy equivalence between the spaces .
[7] These topological spaces, are useful in homotopy theory because it can be used to form a topological space and a model for the homotopy type of the set of homotopy classes of maps
This is because is the set of path components in , that is, there is an
isomorphism of sets
where is the homotopy equivalence.
Fréchet differentiable functions
Let X and Y be two
Banach spaces defined over the same
field, and let C m(U, Y) denote the set of all m-continuously
Fréchet-differentiable functions from the open subset U ⊆ X to Y. The compact-open topology is the
initial topology induced by the
seminorms
where D0 f (x) = f (x), for each compact subset K ⊆ U.[
clarification needed]
See also
References