 Compactopen topology

In mathematics, the compactopen topology is a topology defined on the set of continuous maps between two topological spaces. The compactopen topology is one of the commonlyused topologies on function spaces, and is applied in homotopy theory and functional analysis. It was invented by Ralph Fox in 1945[1].
Contents
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 compactopen topology on C(X,Y). (This collection does not always form a base for a topology on C(X,Y).)
Properties
 If * is a onepoint space then one can identify C(*,X) with X, and under this identification the compactopen topology agrees with the topology on X
 If Y is T_{0}, T_{1}, Hausdorff, regular, or Tychonoff, then the compactopen 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 compactopen topology on C(X,Y).
 If Y is a uniform space (in particular, if Y is a metric space), then the compactopen 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 compactopen 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 compactopen 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 compactopen 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 onepoint space.
 If X is compact, and if Y is a metric space with metric d, then the compactopen 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 denote the set of all mcontinuously Fréchetdifferentiable functions from the open subset to Y. The compactopen topology is the initial topology induced by the seminorms
where , for each compact subset .
See also
 Boundedopen topology
References
 Dugundji, J. (1966), Topology, Allyn and Becon, ISBN B000KWE22K
 O.Ya. Viro, O.A. Ivanov, V.M. Kharlamov and N.Yu. Netsvetaev (2007) Textbook in Problems on Elementary Topology.
 compactopen topology on PlanetMath
Categories: General topology
 Topology of function spaces
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