compact open topology
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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the compact-open topology is a
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
defined on the
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
of continuous maps between two
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
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 vect ...
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 (e.g. inner product, norm, topology, etc.) and the linear functions defi ...
. It was introduced by Ralph Fox in 1945. If the codomain of the functions under consideration has a uniform structure or a metric structure then the compact-open topology is the "topology of
uniform convergence In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions (f_n) converges uniformly to a limiting function f on a set E if, given any arbitrarily ...
on compact sets." 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 called ...
of functions converges in the compact-open topology precisely when it converges uniformly on every compact subset of the
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined ** Domain of definition of a partial function ** Natural domain of a partial function **Domain of holomorphy of a function * ...
.


Definition

Let and be two
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
s, and let denote the set of all continuous maps between and . Given a compact subset of and an
open subset In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are suff ...
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 space In topology, a compactly generated space is a topological space whose topology is coherent with the family of all compact subspaces. Specifically, a topological space ''X'' is compactly generated if it satisfies the following condition: :A subsp ...
s, 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 * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Britis ...
Hausdorff space In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the ma ...
. 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 In category theory, a category is Cartesian closed if, roughly speaking, any morphism defined on a product of two objects can be naturally identified with a morphism defined on one of the factors. These categories are particularly important in mat ...
, 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 * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Britis ...
. 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 top ...
, then can be identified with the
cartesian product In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\t ...
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-seem ...
. * If is , , Hausdorff, regular, or Tychonoff, then the compact-open topology has the corresponding
separation axiom In 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. These are sometim ...
. * 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 together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setti ...
(or more generally, a uniform space), 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 In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions (f_n) converges uniformly to a limiting function f on a set E if, given any arbitrarily ...
. * If and are topological spaces, with
locally compact Hausdorff In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ...
(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-seem ...
). *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 Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathe ...
, then the compact-open topology on is
metrisable In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \inft ...
, and a metric for it is given by for in .


Applications

The compact open topology can be used to topologize the following sets: * \Omega(X,x_0) = \, the
loop space In topology, a branch of mathematics, the loop space Ω''X'' of a pointed topological space ''X'' is the space of (based) loops in ''X'', i.e. continuous pointed maps from the pointed circle ''S''1 to ''X'', equipped with the compact-open topol ...
of X at x_0, * E(X, x_0, x_1) = \, * E(X, x_0) = \. In addition, there is a
homotopy equivalence In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a defo ...
between the spaces C(\Sigma X, Y) \cong C(X, \Omega Y). These topological spaces, C(X,Y) are useful in homotopy theory because it 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 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 them. The word i ...
of sets :\pi(X,Y) \to C(I, C(X, Y))/\sim where \sim is the homotopy equivalence.


Fréchet differentiable functions

Let and be two Banach spaces 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 a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk a ...
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