Compact-open topology


Compact-open topology

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 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 compact-open topology on C(X,Y). (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 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 sequencen} converges to ƒ in the compact-open topology if and only if for every compact subset K of X, {ƒn} 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).

Fréchet differentiable functions

Let X and Y be two Banach spaces defined on the same field, and let \mathcal{C}^m\left(U,Y\right) denote the set of all m-continuously Fréchet-differentiable functions from the open subset U\subseteq X to Y. The compact-open topology is the initial topology induced by the seminorms

p_{K}\left( f \right) = \sup \{ \| D^{j}f\left( x \right)\|, x\in K, 0\leq j \leq m \}

where D^{0}f\left( x \right) = f\left( x \right), for each compact subset K\subseteq U.

See also

  • Bounded-open topology

References

  • Dugundji, J. (1966), Topology, Allyn and Becon, ISBN B000-KWE22-K 

Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Compact complement topology — In mathematics, the compact complement topology is a topology defined on the set of real numbers, defined by declaring a subset open if and only if it is either empty or its complement is compact in the standard Euclidean topology on …   Wikipedia

  • Compact convergence — In mathematics compact convergence (or uniform convergence on compact sets) is a type of convergence which generalizes the idea of uniform convergence. It is associated with the compact open topology. Contents 1 Definition 2 Examples 3 Properties …   Wikipedia

  • Topology — (Greek topos , place, and logos , study ) is the branch of mathematics that studies the properties of a space that are preserved under continuous deformations. Topology grew out of geometry, but unlike geometry, topology is not concerned with… …   Wikipedia

  • Compact space — Compactness redirects here. For the concept in first order logic, see compactness theorem. In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness… …   Wikipedia

  • topology — topologic /top euh loj ik/, topological, adj. topologically, adv. topologist, n. /teuh pol euh jee/, n., pl. topologies for 3. Math. 1. the study of those properties of geometric forms that remain invariant under c …   Universalium

  • Open and closed maps — In topology, an open map is a function between two topological spaces which maps open sets to open sets.[1] That is, a function f : X → Y is open if for any open set U in X, the image f(U) is open in Y. Likewise, a closed map is a function… …   Wikipedia

  • Open book decomposition — In mathematics, an open book decomposition (or simply an open book) is a decomposition of a closed oriented 3 manifold M into a union of surfaces (necessarily with boundary) and solid tori. Open books have relevance to contact geometry, with a… …   Wikipedia

  • Glossary of topology — This is a glossary of some terms used in the branch of mathematics known as topology. Although there is no absolute distinction between different areas of topology, the focus here is on general topology. The following definitions are also… …   Wikipedia

  • List of general topology topics — This is a list of general topology topics, by Wikipedia page. Contents 1 Basic concepts 2 Limits 3 Topological properties 3.1 Compactness and countability …   Wikipedia

  • List of examples in general topology — This is a list of useful examples in general topology, a field of mathematics.* Alexandrov topology * Cantor space * Co kappa topology ** Cocountable topology ** Cofinite topology * Compact open topology * Compactification * Discrete topology *… …   Wikipedia


Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.