compact

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.

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 in C(X, Y) such that f(K) is contained in U. Then the collection of all such V(K, U) is a subbase for the compact-open topology. (This collection does not always form a base for a topology on C(X, Y).)

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, or regular, 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 {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. 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 (f, g) mapsto fog, 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(YZ) × Y → Z, 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 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(f, g) = sup{d(f(x), g(x)) : x in X}, for f, g in C(X, Y).

References

  • Dugundji, James (1966). Topology. Boston, Massachusetts: Allyn and Bacon. ISBN B000-KWE22-K.
  • O.Ya. Viro, O.A. Ivanov, V.M. Kharlamov and N.Yu. Netsvetaev; Textbook in Problems on Elementary Topology; online version

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