View source for Compactification (mathematics)

{{short description|Embedding a topological space into a compact space as a dense subset}}
In [[mathematics]], in [[general topology]], '''compactification''' is the process or result of making a [[topological space]] into a [[compact space]].<ref>{{cite book | last=Munkres | first=James R. | author-link=James Munkres | title=Topology | edition=2nd | publisher=[[Prentice Hall]] | year=2000 | isbn=0-13-181629-2}}</ref> A compact space is a space in which every [[open cover]] of the space contains a finite subcover. 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 line]] 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 is [[homeomorphism|homeomorphic]] to a circle in the plane (which, as a closed and bounded subset of the Euclidean plane, is compact). Every sequence that ran off to infinity in the real line will then converge to ∞ in this compactification.  The direction in which a number approaches infinity on the number line (either in the - direction or + direction) is still preserved on the circle; for if a number approaches towards infinity from the - direction on the number line, then the corresponding point on the circle can approach ∞ from the left for example. Then if a number approaches infinity from the + direction on the number line, then the corresponding point on the circle can approach ∞ from the right.

Intuitively, the process can be pictured as follows: first shrink the real line to the [[open interval]] {{nowrap|(−[[pi|{{pi}}]], {{pi}})}} 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 −{{pi}} to {{pi}} for simplicity.  Identify each such point ''θ'' on the circle with the corresponding point on the real line [[tangent|tan]](''θ''/2).  This function is undefined at the point {{pi}}, since tan({{pi}}/2) is undefined; we will identify this point with our point&nbsp;∞.

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]].

== Definition ==

An [[embedding]] of a topological space ''X'' as a [[dense set|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.

=== Alexandroff one-point compactification ===
{{main|One-point compactification}}
For any noncompact topological space ''X'' the ('''Alexandroff''') '''one-point compactification''' α''X'' of ''X'' is obtained by adding one extra point ∞ (often called a ''point at infinity'') and defining the [[open set]]s of the new space to be the open sets of ''X'' together with the sets of the form ''G''&nbsp;∪&nbsp;{{mset|∞}}, where ''G'' is an open subset of ''X'' such that <math>X \setminus G</math> is closed and compact. The one-point compactification of ''X'' is Hausdorff if and only if ''X'' is Hausdorff and [[locally compact]].<ref>{{citation|first=Pavel S.|last= Alexandroff|author-link=Pavel Alexandroff| journal=  [[Mathematische Annalen]] |volume= 92|issue=3–4 |year=1924|pages= 294–301|title= Über die Metrisation der im Kleinen kompakten topologischen Räume | url=https://eudml.org/doc/159072 | doi=10.1007/BF01448011 | jfm=50.0128.04 }}</ref>

=== Stone–Čech compactification ===
{{main|Stone–Čech compactification}}
Of particular interest are Hausdorff compactifications, i.e., compactifications in which the compact space is [[Hausdorff space|Hausdorff]]. A topological space has a Hausdorff compactification if and only if it is [[Tychonoff space|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 (mathematics)|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 (topology)|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 [[subspace topology|induced topology]] on ''X'' by ''βX'' is the same as the given topology on ''X'', and for any continuous map {{nowrap|''f'' : ''X'' → ''K''}}, where ''K'' is a compact Hausdorff space, there is a unique continuous map {{nowrap|''g'' : ''βX'' → ''K''}} for which ''g'' restricted to ''X'' is identically&nbsp;''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 {{nowrap|[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 {{nowrap|[0, 1]<sup>''C''</sup>}}, the space of ''all'' functions from ''C'' to {{nowrap|[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.<ref>{{cite journal|first=Eduard|last= Čech|author-link=Eduard Čech| title=On bicompact spaces|journal=  [[Annals of Mathematics]] |volume= 38  |year=1937  |pages= 823–844|doi=10.2307/1968839|issue=4|jstor=1968839|hdl= 10338.dmlcz/100420|hdl-access=free}}</ref>
<ref>{{citation|first=Marshall H.|last= Stone|author-link=Marshall H. Stone|title=Applications of the theory of Boolean rings to general topology  |journal=[[Transactions of the American Mathematical Society]] |volume= 41  |year=1937|pages= 375–481
|issue=3|doi=10.2307/1989788 |jstor=1989788|doi-access=free}}</ref>

=== Spacetime compactification ===
[[Walter Benz]] and [[Isaak Yaglom]] have shown how [[stereographic projection]] onto a single-sheet [[hyperboloid]] can be used to provide a [[Motor variable#Compactification|compactification for split complex numbers]]. In fact, the hyperboloid is part of a [[quadric]] in real projective four-space. The method is similar to that used to provide a base manifold for [[group action (mathematics)|group action]] of the [[conformal group of spacetime]].<ref>15 parameter conformal group of spacetime described in {{Wikibooks-inline|Associative Composition Algebra/Homographies}}</ref>

== Projective space ==
[[Real projective space]] '''RP'''<sup>''n''</sup> is a compactification of Euclidean space '''R'''<sup>''n''</sup>. For each possible "direction" in which points in '''R'''<sup>''n''</sup> 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 '''RP'''<sup>1</sup>. Note however that the [[projective plane]] '''RP'''<sup>2</sup> is ''not'' the one-point compactification of the plane '''R'''<sup>2</sup> since more than one point is added.

[[Complex projective space]] '''CP'''<sup>''n''</sup> is also a compactification of  '''C'''<sup>''n''</sup>; the Alexandroff one-point compactification of the plane '''C''' is (homeomorphic to) the complex projective line '''CP'''<sup>1</sup>, 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 '''RP'''<sup>2</sup> intersect in precisely one point, a statement that is not true in '''R'''<sup>2</sup>. 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 of algebraic curves|moduli space of algebraic curves]].

== Compactification and discrete subgroups of Lie groups ==

In the study of [[discrete space|discrete]] subgroups of [[Lie group]]s, the [[Quotient space (topology)|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 (singularity)|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 [[lattice (group)|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 {{math|''n''}}-dimensional [[Euclidean space]] the same questions can be posed, for example about <math>\text{SO}(n) \setminus \text{SL}_n(\textbf{R}) / \text{SL}_n(\textbf{Z}).</math> 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 compactification]]s, that can be formed.

== Other compactification theories ==

* The theories of [[end (topology)|ends of a space]] and [[prime end]]s.
* Some 'boundary' theories such as the [[collaring of an open manifold]], [[Martin boundary]], [[Shilov boundary]] and [[Furstenberg boundary]].
* The [[Bohr compactification]] of a [[topological group]] arises from the consideration of [[almost periodic function]]s.
* The [[projective line over a ring]] for a [[topological ring]] may compactify it.
* The [[Baily–Borel compactification]] of a quotient of a [[Hermitian symmetric space]].
* The [[wonderful compactification]] of a quotient of algebraic groups.
* The compactifications that are simultaneously convex subsets in a locally convex space are called [[convex compactification]]s, their additional linear structure allowing e.g. for developing a differential calculus and more advanced considerations e.g. in relaxation in variational calculus or optimization theory.<ref>{{cite book | last=Roubíček | first=T. | author-link=Tomas Roubicek | title=Relaxation in Optimization Theory and Variational Calculus | publisher=[[W. de Gruyter]] |place = Berlin | year=1997 | isbn=3-11-014542-1}}</ref>

== See also ==

* {{annotated link|Alexandroff extension}}

== References ==
{{reflist}}

[[Category:Compactification (mathematics)| ]]