Compact-open topology

Encyclopedia

In mathematics

, the

defined on the set of continuous maps

between two topological space

s. The compact-open topology is one of the commonly-used topologies on function space

s, and is applied in homotopy theory and functional analysis

. It was invented by Ralph Fox

in 1945http://www.ams.org/journals/bull/1945-51-06/S0002-9904-1945-08370-0/.

s, and let

for the compact-open topology on

for a topology on

s defined on the same field, and let denote the set of all m-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 .

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the

**compact-open topology**is a topologyTopological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...

defined on the set of continuous maps

Continuous function

In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...

between two topological space

Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...

s. The compact-open topology is one of the commonly-used topologies on function space

Function space

In mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in many applications it is a topological space, a vector space, or both.-Examples:...

s, and is applied in homotopy theory and functional analysis

Functional analysis

Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...

. It was invented by Ralph Fox

Ralph Fox

Ralph Hartzler Fox was an American mathematician. As a professor at Princeton University, he taught and advised many of the contributors to the Golden Age of differential topology, and he played an important role in the modernization and main-streaming of knot theory.Ralph Fox attended Swarthmore...

in 1945http://www.ams.org/journals/bull/1945-51-06/S0002-9904-1945-08370-0/.

## Definition

Let*X*and*Y*be two topological spaceTopological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...

s, 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 subsetOpen set

The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...

*U*of*Y*, let*V*(*K*,*U*) denote the set of all functions such that Then the collection of all such*V*(*K*,*U*) is a subbaseSubbase

In topology, a subbase for a topological space X with topology T is a subcollection B of T which generates T, in the sense that T is the smallest topology containing B...

for the compact-open topology on

*C*(*X*,*Y*). (This collection does not always form a baseBase (topology)

In mathematics, a base B for a topological space X with topology T is a collection of open sets in T such that every open set in T can be written as a union of elements of B. We say that the base generates the topology T...

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*T*_{0},*T*_{1}T1 spaceIn topology and related branches of mathematics, a T1 space is a topological space in which, for every pair of distinct points, each has an open neighborhood not containing the other. An R0 space is one in which this holds for every pair of topologically distinguishable points...

, HausdorffHausdorff spaceIn topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...

, regularRegular spaceIn topology and related fields of mathematics, a topological space X is called a regular space if every non-empty closed subset C of X and a point p not contained in C admit non-overlapping open neighborhoods. Thus p and C...

, or TychonoffTychonoff spaceIn topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are kinds of topological spaces.These conditions are examples of separation axioms....

, then the compact-open topology has the corresponding separation axiomSeparation axiomIn topology and related fields of mathematics, there are several restrictions that one often makes on the kinds of topological spaces that one wishes to consider. Some of these restrictions are given by the separation axioms...

.

- If
*X*is Hausdorff and*S*is a subbaseSubbaseIn topology, a subbase for a topological space X with topology T is a subcollection B of T which generates T, in the sense that T is the smallest topology containing B...

for*Y*, then the collection {*V*(*K*,*U*) :*U*in*S*} is a subbaseSubbaseIn topology, a subbase for a topological space X with topology T is a subcollection B of T which generates T, in the sense that T is the smallest topology containing B...

for the compact-open topology on*C*(*X*,*Y*).

- If
*Y*is a uniform spaceUniform spaceIn the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure which is used to define uniform properties such as completeness, uniform continuity and uniform convergence.The conceptual difference between...

(in particular, if*Y*is a metric spaceMetric spaceIn mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean 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 sequenceSequenceIn mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...

{ƒ_{n}} convergeLimit (mathematics)In mathematics, the concept of a "limit" is used to describe the value that a function or sequence "approaches" as the input or index approaches some value. The concept of limit allows mathematicians to define a new point from a Cauchy sequence of previously defined points within a complete metric...

s 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 mapFunction compositionIn mathematics, function composition is the application of one function to the results of another. For instance, the functions and can be composed by computing the output of g when it has an argument of f instead of x...

given by is continuous (here all the function spaces are given the compact-open topology and*C*(*Y*,*Z*) ×*C*(*X*,*Y*) is given the product topologyProduct topologyIn topology and related areas of mathematics, a product space is the cartesian product of a family of topological spaces equipped with a natural topology called 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*) = supSupremumIn mathematics, given a subset S of a totally or partially ordered set T, the supremum of S, if it exists, is the least element of T that is greater than or equal to every element of S. Consequently, the supremum is also referred to as the least upper bound . If the supremum exists, it is unique...

{*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 spaceBanach space

In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...

s defined on the same field, and let denote the set of all m-continuously Fréchet-differentiable functions from the open subset to . The compact-open topology is the initial topology

Initial topology

In general topology and related areas of mathematics, the initial topology on a set X, with respect to a family of functions on X, is the coarsest topology on X which makes those functions continuous.The subspace topology and product topology constructions are both special cases of initial...

induced by the seminorms

where , for each compact subset .