Let and be two topological spaces, and let denote the set of all continuous maps between and. Given a compact subset of and an open subset of, let denote the set of all functions such that Then the collection of all such is a subbase for the compact-open topology on. When working in the category of compactly generated spaces, it is common to modify this definition by restricting to the subbase formed from those that are the image of a compact Hausdorff space. Of course, if 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 with, and under this identification the compact-open topology agrees with the topology on. More generally, if is a discrete space, then can be identified with the cartesian product of copies of and the compact-open topology agrees with the product topology.
If is T0 space|, T1 space|, Hausdorff, regular, or Tychonoff, then the compact-open topology has the corresponding separation axiom.
If is Hausdorff and is a subbase for, then the collection is a subbase for the compact-open topology on.
If is a locally compact Hausdorff space, then the evaluation map, defined by, is continuous. This can be seen as a special case of the above where is a one-point space.
If is compact, and is a metric space with metric, then the compact-open topology on is metrisable, and a metric for it is given by for.
Applications
The compact open topology can be used to topologize the following sets
In addition, there is a homotopy equivalence between the spaces. 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 mapsThis is because is the set of path components in, that is, there is an isomorphism of setswhere is the homotopy equivalence.
Let and be two Banach spaces defined over the same field, and let denote the set of all -continuously Fréchet-differentiable functions from the open subset to. The compact-open topology is the initial topology induced by the seminorms where, for each compact subset.
