Compactification (mathematics)
Encyclopedia
In mathematics
, compactification is the process or result of making a topological space
compact
. The methods of compactification are various, but each is a way of controlling points from "going off to infinity" by in some way adding "points at infinity" or preventing such an "escape".
with its ordinary topology. This space is not compact; in a sense, points can go off to infinity to the left or to the right. It is possible to turn the real line into a compact space by adding a single "point at infinity" which we will denote by ∞. The resulting compactification can be thought of as a circle (which is compact as a closed and bounded subset of the Euclidean plane). Every sequence that ran off to infinity in the real line will then converge to ∞ in this compactification.
Intuitively, the process can be pictured as follows: first shrink the real line to the open interval (-π
,π) on the x-axis; then bend the ends of this interval upwards (in positive y-direction) and move them towards each other, until you get a circle with one point (the topmost one) missing. This point is our new point ∞ "at infinity"; adding it in completes the compact circle.
A bit more formally: we represent a point on the unit circle
by its angle
, in radian
s, going from -π to π for simplicity. Identify each such point θ on the circle with the corresponding point on the real line tan
(θ/2). This function is undefined at the point π, since tan(π/2) is undefined; we will identify this point with our point ∞.
Since tangents and inverse tangents are both continuous, our identification function is a homeomorphism
between the real line and the unit circle without ∞. What we have constructed is called the Alexandroff one-point compactification of the real line, discussed in more generality below. It is also possible to compactify the real line by adding two points, +∞ and -∞; this results in the extended real line.
of a topological space X as a dense
subset of a compact space is called a compactification of X. It is often useful to embed topological space
s in compact space
s, because of the special properties compact spaces have.
Embeddings into compact Hausdorff space
s may be of particular interest. Since every compact Hausdorff space is a Tychonoff space
, and every subspace of a Tychonoff space is Tychonoff, we conclude that any space possessing a Hausdorff compactification must be a Tychonoff space. In fact, the converse is also true; being a Tychonoff space is both necessary and sufficient for possessing a Hausdorff compactification.
The fact that large and interesting classes of non-compact spaces do in fact have compactifications of particular sorts makes compactification a common technique in topology.
s of the new space to be the open sets of X together with the sets of the form G U {∞}, where G is an open subset of X such that X \ G is closed and compact. The one-point compactification of X is Hausdorff if and only if X is Hausdorff and locally compact.
. A topological space has a Hausdorff compactification if and only if it is Tychonoff
. In this case, there is a unique (up to
homeomorphism
) "most general" Hausdorff compactification, the Stone–Čech compactification
of X, denoted by βX; formally, this exhibits the category
of Compact Hausdorff spaces and continuous maps as a reflective subcategory
of the category of Tychonoff spaces and continuous maps.
"Most general" or formally "reflective" means that the space βX is characterized by the universal property
that any continuous function from X to a compact Hausdorff space K can be extended to a continuous function from βX to K in a unique way. More explicitly, βX is a compact Hausdorff space containing X such that the induced topology
on X by βX is the same as the given topology on X, and for any continuous map f:X → K, where K is a compact Hausdorff space, there is a unique continuous map g:βX → K for which g restricted to X is identically f.
The Stone–Čech compactification can be constructed explicitly as follows: let C be the set of continuous functions from X to the closed interval [0,1]. Then each point in X can be identified with an evaluation function on C. Thus X can be identified with a subset of [0,1]C, the space of all functions from C to [0,1]. Since the latter is compact by Tychonoff's theorem
, the closure of X as a subset of that space will also be compact. This is the Stone–Čech compactification.
RPn is a compactification of Euclidean space Rn. For each possible "direction" in which points in Rn can "escape", one new point at infinity is added (but each direction is identified with its opposite). The Alexandroff one-point compactification of R we constructed in the example above is in fact homeomorphic to RP1. Note however that the projective plane
RP2 is not the one-point compactification of the plane R2 since more than one point is added.
Complex projective space
CPn is also a compactification of Cn; the Alexandroff one-point compactification of the plane C is (homeomorphic to) the complex projective line CP1, which in turn can be identified with a sphere, the Riemann sphere
.
Passing to projective space is a common tool in algebraic geometry
because the added points at infinity lead to simpler formulations of many theorems. For example, any two different lines in RP2 intersect in precisely one point, a statement that is not true in R2. More generally, Bézout's theorem
, which is fundamental in intersection theory, holds in projective space but not affine space. This distinct behavior of intersections in affine space and projective space is reflected in algebraic topology
in the cohomology ring
s – the cohomology of affine space is trivial, while the cohomology of projective space is non-trivial and reflects the key features of intersection theory (dimension and degree of a subvariety, with intersection being Poincaré dual to the cup product
).
Compactification of moduli space
s generally require allowing certain degeneracies – for example, allowing certain singularities or reducible varieties. This is notably used in the Deligne–Mumford compactification of the moduli space of algebraic curves.
subgroups of Lie group
s, the quotient space
of coset
s is often a candidate for more subtle compactification to preserve structure at a richer level than just topological.
For example modular curve
s are compactified by the addition of single points for each cusp
, making them Riemann surface
s (and so, since they are compact, algebraic curve
s). Here the cusps are there for a good reason: the curves parametrize a space of lattices
, and those lattices can degenerate ('go off to infinity'), often in a number of ways (taking into account some auxiliary structure of level). The cusps stand in for those different 'directions to infinity'.
That is all for lattices in the plane. In n-dimensional Euclidean space
the same questions can be posed, for example about SO(n)\SLn(R)/SLn(Z). This is harder to compactify. There are a variety of compactifications, such as the Borel-Serre compactification, the reductive Borel-Serre compactification, and the Satake compactifications, that can be formed.
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...
, compactification is the process or result of making a 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...
compact
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
. The methods of compactification are various, but each is a way of controlling points from "going off to infinity" by in some way adding "points at infinity" or preventing such an "escape".
An example
Consider the real lineReal line
In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one...
with its ordinary topology. This space is not compact; in a sense, points can go off to infinity to the left or to the right. It is possible to turn the real line into a compact space by adding a single "point at infinity" which we will denote by ∞. The resulting compactification can be thought of as a circle (which is compact as a closed and bounded subset of the Euclidean plane). Every sequence that ran off to infinity in the real line will then converge to ∞ in this compactification.
Intuitively, the process can be pictured as follows: first shrink the real line to the open interval (-π
Pi
' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...
,π) on the x-axis; then bend the ends of this interval upwards (in positive y-direction) and move them towards each other, until you get a circle with one point (the topmost one) missing. This point is our new point ∞ "at infinity"; adding it in completes the compact circle.
A bit more formally: we represent a point on the unit circle
Unit circle
In mathematics, a unit circle is a circle with a radius of one. Frequently, especially in trigonometry, "the" unit circle is the circle of radius one centered at the origin in the Cartesian coordinate system in the Euclidean plane...
by its angle
Angle
In geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...
, in radian
Radian
Radian is the ratio between the length of an arc and its radius. The radian is the standard unit of angular measure, used in many areas of mathematics. The unit was formerly a SI supplementary unit, but this category was abolished in 1995 and the radian is now considered a SI derived unit...
s, going from -π to π for simplicity. Identify each such point θ on the circle with the corresponding point on the real line tan
Tangent
In geometry, the tangent line to a plane curve at a given point is the straight line that "just touches" the curve at that point. More precisely, a straight line is said to be a tangent of a curve at a point on the curve if the line passes through the point on the curve and has slope where f...
(θ/2). This function is undefined at the point π, since tan(π/2) is undefined; we will identify this point with our point ∞.
Since tangents and inverse tangents are both continuous, our identification function is a homeomorphism
Homeomorphism
In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are...
between the real line and the unit circle without ∞. What we have constructed is called the Alexandroff one-point compactification of the real line, discussed in more generality below. It is also possible to compactify the real line by adding two points, +∞ and -∞; this results in the extended real line.
Definition
An embeddingEmbedding
In mathematics, an embedding is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup....
of a topological space X as a dense
Dense set
In topology and related areas of mathematics, a subset A of a topological space X is called dense if any point x in X belongs to A or is a limit point of A...
subset of a compact space is called a compactification of X. It is often useful to embed 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 in compact space
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
s, because of the special properties compact spaces have.
Embeddings into compact Hausdorff space
Hausdorff space
In 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...
s may be of particular interest. Since every compact Hausdorff space is a Tychonoff space
Tychonoff space
In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are kinds of topological spaces.These conditions are examples of separation axioms....
, and every subspace of a Tychonoff space is Tychonoff, we conclude that any space possessing a Hausdorff compactification must be a Tychonoff space. In fact, the converse is also true; being a Tychonoff space is both necessary and sufficient for possessing a Hausdorff compactification.
The fact that large and interesting classes of non-compact spaces do in fact have compactifications of particular sorts makes compactification a common technique in topology.
Alexandroff one-point compactification
For any topological space X the (Alexandroff) one-point compactification αX of X is obtained by adding one extra point ∞ (often called a point at infinity) and defining the open setOpen 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...
s of the new space to be the open sets of X together with the sets of the form G U {∞}, where G is an open subset of X such that X \ G is closed and compact. The one-point compactification of X is Hausdorff if and only if X is Hausdorff and locally compact.
Stone–Čech compactification
Of particular interest are Hausdorff compactifications, i.e., compactifications in which the compact space is HausdorffHausdorff space
In 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...
. A topological space has a Hausdorff compactification if and only if it is Tychonoff
Tychonoff space
In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are kinds of topological spaces.These conditions are examples of separation axioms....
. In this case, there is a unique (up to
Up to
In mathematics, the phrase "up to x" means "disregarding a possible difference in x".For instance, when calculating an indefinite integral, one could say that the solution is f "up to addition by a constant," meaning it differs from f, if at all, only by some constant.It indicates that...
homeomorphism
Homeomorphism
In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are...
) "most general" Hausdorff compactification, the Stone–Čech compactification
Stone–Cech compactification
In the mathematical discipline of general topology, Stone–Čech compactification is a technique for constructing a universal map from a topological space X to a compact Hausdorff space βX...
of X, denoted by βX; formally, this exhibits the category
Category (mathematics)
In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...
of Compact Hausdorff spaces and continuous maps as a reflective subcategory
Reflective subcategory
In mathematics, a subcategory A of a category B is said to be reflective in B when the inclusion functor from A to B has a left adjoint. This adjoint is sometimes called a reflector...
of the category of Tychonoff spaces and continuous maps.
"Most general" or formally "reflective" means that the space βX is characterized by the universal property
Universal property
In various branches of mathematics, a useful construction is often viewed as the “most efficient solution” to a certain problem. The definition of a universal property uses the language of category theory to make this notion precise and to study it abstractly.This article gives a general treatment...
that any continuous function from X to a compact Hausdorff space K can be extended to a continuous function from βX to K in a unique way. More explicitly, βX is a compact Hausdorff space containing X such that the induced topology
Subspace topology
In topology and related areas of mathematics, a subspace of a topological space X is a subset S of X which is equipped with a natural topology induced from that of X called the subspace topology .- Definition :Given a topological space and a subset S of X, the...
on X by βX is the same as the given topology on X, and for any continuous map f:X → K, where K is a compact Hausdorff space, there is a unique continuous map g:βX → K for which g restricted to X is identically f.
The Stone–Čech compactification can be constructed explicitly as follows: let C be the set of continuous functions from X to the closed interval [0,1]. Then each point in X can be identified with an evaluation function on C. Thus X can be identified with a subset of [0,1]C, the space of all functions from C to [0,1]. Since the latter is compact by Tychonoff's theorem
Tychonoff's theorem
In mathematics, Tychonoff's theorem states that the product of any collection of compact topological spaces is compact. The theorem is named after Andrey Nikolayevich Tychonoff, who proved it first in 1930 for powers of the closed unit interval and in 1935 stated the full theorem along with the...
, the closure of X as a subset of that space will also be compact. This is the Stone–Čech compactification.
Projective space
Real projective spaceReal projective space
In mathematics, real projective space, or RPn, is the topological space of lines through 0 in Rn+1. It is a compact, smooth manifold of dimension n, and a special case of a Grassmannian.-Construction:...
RPn is a compactification of Euclidean space Rn. For each possible "direction" in which points in Rn can "escape", one new point at infinity is added (but each direction is identified with its opposite). The Alexandroff one-point compactification of R we constructed in the example above is in fact homeomorphic to RP1. Note however that the projective plane
Projective plane
In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines that do not intersect...
RP2 is not the one-point compactification of the plane R2 since more than one point is added.
Complex projective space
Complex projective space
In mathematics, complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space label the lines through the origin of a real Euclidean space, the points of a complex projective space label the complex lines...
CPn is also a compactification of Cn; the Alexandroff one-point compactification of the plane C is (homeomorphic to) the complex projective line CP1, which in turn can be identified with a sphere, the Riemann sphere
Riemann sphere
In mathematics, the Riemann sphere , named after the 19th century mathematician Bernhard Riemann, is the sphere obtained from the complex plane by adding a point at infinity...
.
Passing to projective space is a common tool in algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
because the added points at infinity lead to simpler formulations of many theorems. For example, any two different lines in RP2 intersect in precisely one point, a statement that is not true in R2. More generally, Bézout's theorem
Bézout's theorem
Bézout's theorem is a statement in algebraic geometry concerning the number of common points, or intersection points, of two plane algebraic curves. The theorem claims that the number of common points of two such curves X and Y is equal to the product of their degrees...
, which is fundamental in intersection theory, holds in projective space but not affine space. This distinct behavior of intersections in affine space and projective space is reflected in algebraic topology
Algebraic topology
Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although algebraic topology...
in the cohomology ring
Cohomology ring
In mathematics, specifically algebraic topology, the cohomology ring of a topological space X is a ring formed from the cohomology groups of X together with the cup product serving as the ring multiplication. Here 'cohomology' is usually understood as singular cohomology, but the ring structure is...
s – the cohomology of affine space is trivial, while the cohomology of projective space is non-trivial and reflects the key features of intersection theory (dimension and degree of a subvariety, with intersection being Poincaré dual to the cup product
Cup product
In mathematics, specifically in algebraic topology, the cup product is a method of adjoining two cocycles of degree p and q to form a composite cocycle of degree p + q. This defines an associative graded commutative product operation in cohomology, turning the cohomology of a space X into a...
).
Compactification of moduli space
Moduli space
In algebraic geometry, a moduli space is a geometric space whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects...
s generally require allowing certain degeneracies – for example, allowing certain singularities or reducible varieties. This is notably used in the Deligne–Mumford compactification of the moduli space of algebraic curves.
Compactification and discrete subgroups of Lie groups
In the study of discreteDiscrete space
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are "isolated" from each other in a certain sense.- Definitions :Given a set X:...
subgroups of Lie group
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
s, the quotient space
Quotient space
In topology and related areas of mathematics, a quotient space is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space. The points to be identified are specified by an equivalence relation...
of coset
Coset
In mathematics, if G is a group, and H is a subgroup of G, and g is an element of G, thenA coset is a left or right coset of some subgroup in G...
s is often a candidate for more subtle compactification to preserve structure at a richer level than just topological.
For example modular curve
Modular curve
In number theory and algebraic geometry, a modular curve Y is a Riemann surface, or the corresponding algebraic curve, constructed as a quotient of the complex upper half-plane H by the action of a congruence subgroup Γ of the modular group of integral 2×2 matrices SL...
s are compactified by the addition of single points for each cusp
Cusp (singularity)
In the mathematical theory of singularities a cusp is a type of singular point of a curve. Cusps are local singularities in that they are not formed by self intersection points of the curve....
, making them Riemann surface
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the...
s (and so, since they are compact, algebraic curve
Algebraic curve
In algebraic geometry, an algebraic curve is an algebraic variety of dimension one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections.- Plane algebraic curves...
s). Here the cusps are there for a good reason: the curves parametrize a space of lattices
Lattice (group)
In mathematics, especially in geometry and group theory, a lattice in Rn is a discrete subgroup of Rn which spans the real vector space Rn. Every lattice in Rn can be generated from a basis for the vector space by forming all linear combinations with integer coefficients...
, and those lattices can degenerate ('go off to infinity'), often in a number of ways (taking into account some auxiliary structure of level). The cusps stand in for those different 'directions to infinity'.
That is all for lattices in the plane. In n-dimensional Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
the same questions can be posed, for example about SO(n)\SLn(R)/SLn(Z). This is harder to compactify. There are a variety of compactifications, such as the Borel-Serre compactification, the reductive Borel-Serre compactification, and the Satake compactifications, that can be formed.
Other compactification theories
- The theories of ends of a space and prime endPrime endIn mathematics, the prime end compactification is a method to compactify a topological disc by adding a circle in an appropriate way....
s. - Some 'boundary' theories such as the collaring of an open manifold, Martin boundary, Shilov boundaryShilov boundaryIn functional analysis, a branch of mathematics, the Shilov boundary is the smallest closed subset of the structure space of a commutative Banach algebra where an analog of the maximum modulus principle holds...
and Fürstenberg boundary. - The Bohr compactificationBohr compactificationIn mathematics, the Bohr compactification of a topological group G is a compact Hausdorff topological group H that may be canonically associated to G. Its importance lies in the reduction of the theory of uniformly almost periodic functions on G to the theory of continuous functions on H...
of a topological groupTopological groupIn mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology. A topological group is a mathematical object with both an algebraic structure and a...
arises from the consideration of almost periodic functionAlmost periodic functionIn mathematics, an almost periodic function is, loosely speaking, a function of a real number that is periodic to within any desired level of accuracy, given suitably long, well-distributed "almost-periods". The concept was first studied by Harald Bohr and later generalized by Vyacheslav Stepanov,...
s. - One can compactify a topological ringTopological ringIn mathematics, a topological ring is a ring R which is also a topological space such that both the addition and the multiplication are continuous as mapswhere R × R carries the product topology.- General comments :...
by forming a projective line with inversive ring geometryInversive ring geometryIn mathematics, inversive ring geometry is the extension of the concepts of projective line, homogeneous coordinates, projective transformations, and cross-ratio to the context of associative rings, concepts usually built upon rings that happen to be fields....
. - The Baily-Borel compactification of a quotient of a hermitean symmetric space.