Compact-open Topology - Properties

Properties

• If * is a one-point space then one can identify C(*,X) with X, and under this identification the compact-open topology agrees with the topology on X

• 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 in S} is a subbase for the compact-open topology on C(X,Y).

• If Y is a uniform space (in particular, if Y is a metric space), then the compact-open 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 compact-open 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 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 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 compact-open 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 one-point space.

• If X is compact, and if 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(ƒ,g) = sup{d(ƒ(x), g(x)) : x in X}, for ƒ, g in C(X,Y).