compact-open topology
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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the compact-open topology is a
topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
defined on the set of continuous maps between two
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
s. The compact-open topology is one of the commonly used topologies on
function space In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a ve ...
s, and is applied in homotopy theory and
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 (for example, Inner product space#Definition, inner product, Norm (mathematics ...
. It was introduced by Ralph Fox in 1945. If the
codomain In mathematics, a codomain, counter-domain, or set of destination of a function is a set into which all of the output of the function is constrained to fall. It is the set in the notation . The term '' range'' is sometimes ambiguously used to ...
of the functions under consideration has a uniform structure or a metric structure then the compact-open topology is the "topology of uniform convergence on
compact set In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space. The idea is that a compact space has no "punctures" or "missing endpoints", i.e., i ...
s." That is to say, a
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is cal ...
of functions converges in the compact-open topology precisely when it converges uniformly on every compact subset of the domain.


Definition

Let and be two
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
s, and let denote the set of all continuous maps between and . Given a compact subset of and an open subset of , let denote the set of all functions such that In other words, V(K, U) = C(K, U) \times_ C(X, Y). Then the collection of all such is a subbase for the compact-open topology on . (This collection does not always form a base for a topology on .) When working in the category of compactly generated spaces, it is common to modify this definition by restricting to the subbase formed from those that are the image of a
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...
Hausdorff space In topology and related branches of mathematics, a Hausdorff space ( , ), T2 space or separated space, is a topological space where distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topologi ...
. Of course, if is compactly generated and Hausdorff, this definition coincides with the previous one. However, the modified definition is crucial if one wants the convenient category of compactly generated weak Hausdorff spaces to be Cartesian closed, among other useful properties. The confusion between this definition and the one above is caused by differing usage of the word
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...
. If is locally compact, then X \times - from the category of topological spaces always has a right adjoint Hom(X, -) . This adjoint coincides with the compact-open topology and may be used to uniquely define it. The modification of the definition for compactly generated spaces may be viewed as taking the adjoint of the product in the category of compactly generated spaces instead of the category of topological spaces, which ensures that the right adjoint always exists.


Properties

* If is a one-point space then one can identify with , and under this identification the compact-open topology agrees with the topology on . More generally, if is a
discrete space In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are '' isolated'' from each other in a certain sense. The discrete topology is the finest to ...
, then can be identified with the
cartesian product In mathematics, specifically set theory, the Cartesian product of two sets and , denoted , is the set of all ordered pairs where is an element of and is an element of . In terms of set-builder notation, that is A\times B = \. A table c ...
of copies of and the compact-open topology agrees with the
product topology In 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. This topology differs from another, perhaps more natural-seemin ...
. * If is , , Hausdorff, regular, or Tychonoff, then the compact-open topology has the corresponding separation axiom. * If is Hausdorff and is a subbase for , then the collection is a subbase for the compact-open topology on . * If is a
metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
(or more generally, a
uniform space In the mathematical field of topology, a uniform space is a topological space, set with additional mathematical structure, structure that is used to define ''uniform property, uniform properties'', such as complete space, completeness, uniform con ...
), then the compact-open topology is equal to the topology of compact convergence. In other words, if is a metric space, then a sequence converges to in the compact-open topology if and only if for every compact subset of , converges uniformly to on . If is compact and is a uniform space, then the compact-open topology is equal to the topology of uniform convergence. * If and are topological spaces, with locally compact Hausdorff (or even just locally compact preregular), then the composition map given by is continuous (here all the function spaces are given the compact-open topology and is given the
product topology In 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. This topology differs from another, perhaps more natural-seemin ...
). *If is a locally compact Hausdorff (or preregular) space, then the evaluation map , defined by , is continuous. This can be seen as a special case of the above where is a one-point space. * If is compact, and is a metric space with metric , then the compact-open topology on is metrizable, and a metric for it is given by for in . More generally, if is hemicompact, and metric, the compact-open topology is metrizable by the construction linked here.


Applications

The compact open topology can be used to topologize the following sets: * \Omega(X,x_0) = \, the loop space of X at x_0, * E(X, x_0, x_1) = \, * E(X, x_0) = \. In addition, there is a homotopy equivalence between the spaces C(\Sigma X, Y) \cong C(X, \Omega Y). The topological spaces C(X,Y) are useful in homotopy theory because they can be used to form a topological space and a model for the homotopy type of the ''set'' of homotopy classes of maps \pi(X,Y) = \. This is because \pi(X,Y) is the set of path components in C(X,Y)that is, there is an
isomorphism In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
of sets \pi(X,Y) \to C(I, C(X, Y))/, where \sim is the homotopy equivalence.


Fréchet differentiable functions

Let and be two
Banach space In mathematics, more specifically in functional analysis, a Banach space (, ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and ...
s defined over the same field, and let denote the set of all -continuously Fréchet-differentiable functions from the open subset to . The compact-open topology is the initial topology induced by the
seminorm In mathematics, particularly in functional analysis, a seminorm is like a Norm (mathematics), norm but need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some Absorbing ...
s :p_(f) = \sup \left\ where , for each compact subset .


See also

* Topology of uniform convergence *


References

* * O.Ya. Viro, O.A. Ivanov, V.M. Kharlamov and N.Yu. Netsvetaev (2007
Textbook in Problems on Elementary Topology
* {{planetmath reference, urlname=CompactOpenTopology, title=Compact-open topology *

Ronald Brown, 2006 General topology Topology of function spaces