Editing Stone–Čech compactification

{{Short description|Concept in topology}}
In the mathematical discipline of [[general topology]], '''Stone–Čech compactification''' (or '''Čech–Stone compactification'''<ref>M. Henriksen, "Rings of continuous functions in the 1950s", in ''Handbook of the History of General Topology'', edited by C. E. Aull, R. Lowen,
Springer Science & Business Media, 2013, p. 246</ref>) is a technique for constructing a [[Universal property|universal map]] from a [[topological space]] ''X'' to a [[Compact space|compact]] [[Hausdorff space]] ''&beta;X''. The Stone–Čech compactification ''βX'' of a topological  space ''X'' is the largest, most general compact Hausdorff space "generated" by ''X'', in the sense that any continuous map from ''X'' to a compact Hausdorff space [[List of mathematical jargon#factor through|factors through]] ''βX'' (in a unique way). If ''X'' is a [[Tychonoff space]] then the map from ''X'' to its [[image (mathematics)|image]] in ''βX'' is a [[homeomorphism]], so ''X'' can be thought of as a ([[Dense (topology)|dense]]) subspace of ''βX''; every other compact Hausdorff space that densely contains ''X'' is a [[Quotient space (topology)|quotient]] of ''βX''. For general topological spaces ''X'', the map from ''X'' to ''βX'' need not be [[Injective function|injective]].

A form of the [[axiom of choice]] is required to prove that every topological space has a Stone–Čech compactification. Even for quite simple spaces ''X'', an accessible concrete description of ''βX'' often remains elusive. In particular, proofs that {{math|''βX'' &setminus; ''X''}} is nonempty do not give an explicit description of any particular point in {{math|''βX'' &setminus; ''X''}}.

The Stone–Čech compactification occurs implicitly in a paper by  {{harvs|txt|authorlink=Andrey Nikolayevich Tikhonov|last=Tychonoff|first=Andrey Nikolayevich|year=1930}} and was given explicitly by {{harvs|authorlink=Marshall Stone|first=Marshall|last=Stone|year=1937|txt=yes}} and {{harvs|authorlink=Eduard Čech|first=Eduard |last=Čech|year=1937|txt=yes}}.

== History ==

[[Andrey Nikolayevich Tikhonov]] introduced completely regular spaces in 1930 in order to avoid the pathological situation of [[Hausdorff space]]s whose only continuous [[real number|real]]-valued functions are constant maps.{{sfn | Narici|Beckenstein | 2011 | p=240}}

In the same 1930 article where Tychonoff defined completely regular spaces, he also proved that every [[Tychonoff space]] (i.e. [[Hausdorff space|Hausdorff]] completely regular space) has a Hausdorff [[Compactification (mathematics)|compactification]] (in this same article, he also proved [[Tychonoff's theorem]]). 
In 1937, Čech extended Tychonoff's technique and introduced the notation ''βX'' for this compactification. 
Stone also constructed ''βX'' in a 1937 article, although using a very different method. 
Despite Tychonoff's article being the first work on the subject of the Stone–Čech compactification and despite Tychonoff's article being referenced by both Stone and Čech, Tychonoff's name is rarely associated with ''βX''.{{sfn | Narici|Beckenstein | 2011 | pp=225-273}}

== Universal property and functoriality ==

The Stone–Čech compactification of the topological space ''X'' is a compact Hausdorff space  ''βX'' together with a continuous map ''i<sub>X</sub>'' : ''X'' → ''βX'' that has the following [[universal property]]: any [[continuous map]] ''f'' : ''X'' → ''K'', where ''K'' is a compact Hausdorff space, extends uniquely to a continuous map ''βf'' : ''βX'' → ''K'', i.e. ({{itco|''βf''}})''i<sub>X</sub>'' = ''f''.{{sfn|Munkres|2000|pp=239, Theorem 38.4}}
[[File:Stone cech diagram.svg|center|The universal property of the Stone-Cech compactification expressed in diagram form.|250px]]
As is usual for universal properties, this universal property characterizes ''βX'' [[up to]] [[homeomorphism]].

As is outlined in {{sectionlink||Constructions}}, below, one can prove (using the axiom of choice) that such a Stone–Čech compactification ''i<sub>X</sub>'' : ''X'' → ''βX'' exists for every topological space ''X''.  Furthermore, the image ''i<sub>X</sub>''(''X'') is dense in ''βX''.

Some authors add the assumption  that the starting space ''X'' be Tychonoff (or even [[locally compact]] Hausdorff), for the following reasons:
*The map from ''X''  to its image in  ''βX'' is a homeomorphism if and only if ''X'' is Tychonoff. 
*The map from ''X'' to its image  in ''βX'' is a homeomorphism to an open subspace  if and only if ''X'' is locally compact Hausdorff.
The Stone–Čech construction can be performed for more general spaces ''X'', but in that case the map ''X'' → ''βX'' need not be a homeomorphism to the image of ''X''  (and sometimes is not even injective).

As is usual for universal constructions like this, the extension property makes ''β'' a [[functor]] from '''Top''' (the [[category of topological spaces]]) to '''CHaus''' (the category of compact Hausdorff spaces). Further, if we let ''U'' be the [[inclusion functor]] from '''CHaus''' into '''Top''', maps from ''βX'' to ''K'' (for ''K'' in '''CHaus''') correspond [[Bijection|bijectively]] to maps from ''X'' to ''UK'' (by considering their [[Restriction (mathematics)|restriction]] to ''X'' and using the universal property of ''βX''). i.e. 
:Hom(''βX'', ''K'') ≅ Hom(''X'', ''UK''), 
which means that ''β'' is [[adjoint functor|left adjoint]] to ''U''. This implies that '''CHaus''' is a [[reflective subcategory]] of '''Top''' with reflector ''β''.

== Examples ==
If ''X'' is a compact Hausdorff space, then it coincides with its Stone–Čech compactification.{{sfn|Munkres|2000|pp=241}}

The Stone–Čech compactification of the [[first uncountable ordinal]] <math>\omega_1</math>, with the [[order topology]], is the ordinal <math>\omega_1 + 1</math>. The Stone–Čech compactification of the [[deleted Tychonoff plank]] is the Tychonoff plank.<ref>{{Cite book|url=https://books.google.com/books?id=zhP2CAAAQBAJ&pg=PA95|title=The Stone-Čech Compactification|last=Walker|first=R. C.|publisher=Springer|year=1974|isbn=978-3-642-61935-9|pages=95–97|language=en}}</ref>

==Constructions==

===Construction using products===
One attempt to construct the Stone–Čech compactification of ''X'' is to take the closure of the image of ''X'' in

:<math>\prod\nolimits_{f:X\to K} K</math>

where the product is over all maps from ''X'' to compact Hausdorff spaces ''K'' (or, equivalently, the image of ''X'' by the right [[Kan extension]] of the identity functor of the category ''CHaus'' of compact Hausdorff spaces along the inclusion functor of ''CHaus'' into the category ''Top'' of general topological spaces).<ref group=Note>Refer to Example 4.6.12 for an explicit left adjoint construction, or to Proposition 6.5.2 for how left adjoints can be seen as right Kan extensions in {{cite book | author=Riehl | title=Category Theory in Context |year=2014|page=149, 210}}</ref> By [[Tychonoff's theorem]] this product of compact spaces is compact, and the closure of ''X'' in this space is therefore also compact. This works intuitively but fails for the technical reason that  the collection of all such maps is a [[proper class]] rather than a set. There are several ways to modify this idea to make it work; for example, one can restrict the compact Hausdorff spaces ''K'' to have underlying set ''P''(''P''(''X'')) (the [[power set]] of the power set of ''X''), which is sufficiently large that it has cardinality at least equal to that of every compact Hausdorff space to which ''X'' can be mapped with dense image.

===Construction using the unit interval===
One way of constructing ''βX'' is to let ''C'' be the set of all [[continuous function]]s from ''X'' into [0, 1] and consider the map <math> e: X \to [0,1]^{C} </math> where 

:<math> e(x): f \mapsto f(x) </math>

This may be seen to be a continuous map onto its image, if [0, 1]<sup>''C''</sup> is given the [[product topology]]. By [[Tychonoff's theorem]] we have that [0, 1]<sup>''C''</sup> is compact since [0, 1] is. Consequently, the closure of ''X'' in [0, 1]<sup>''C''</sup> is a compactification of ''X''.

In fact, this closure is the Stone–Čech compactification. To verify this, we just need to verify that the closure satisfies the appropriate universal property. We do this first for ''K'' = [0, 1], where the desired extension of ''f'' : ''X'' → [0, 1] is just the projection onto the ''f'' coordinate in [0, 1]<sup>''C''</sup>. In order to then get this for general compact Hausdorff ''K'' we use the above to note that ''K'' can be embedded in some cube, extend each of the coordinate functions and then take the product of these extensions.

The special property of the [[unit interval]] needed for this construction to work is that it is a ''cogenerator'' of the category of compact Hausdorff spaces: this means that if ''A'' and ''B'' are compact Hausdorff spaces, and ''f'' and ''g'' are distinct maps from ''A'' to ''B'', then there is a map ''h'' : ''B'' → [0, 1] such that ''hf'' and ''hg'' are distinct. Any other cogenerator (or cogenerating set) can be used in this construction.

===Construction using ultrafilters===
{{See also|Stone topology|Filters in topology#Stone topology}}

Alternatively, if {{mvar|X}} is [[Discrete space|discrete]], then it is possible to construct <math>\beta X</math> as the set of all [[Ultrafilter (set theory)|ultrafilter]]s on {{mvar|X}}, with the elements of {{mvar|X}} corresponding to the [[Ultrafilter|principal ultrafilter]]s. The topology on the set of ultrafilters, known as the {{em|[[Stone topology|{{visible anchor|Stone topology|Ultrafilter construction}}]]}}, is generated by sets of the form <math>\{ F : U \in F \}</math> for {{mvar|U}} a subset of {{mvar|X}}.

Again we verify the universal property: For <math>f : X \to K</math> with {{mvar|K}} compact Hausdorff and {{mvar|F}} an ultrafilter on {{mvar|X}} we have an [[Filter (set theory)#Ultrafilters|ultrafilter base]] <math>f(F)</math> on {{mvar|K}}, the [[Pushforward (differential)|pushforward]] of {{mvar|F}}. This has a unique [[Limit (mathematics)|limit]] because {{mvar|K}} is compact Hausdorff, say {{mvar|x}}, and we define <math>\beta f(F) = x.</math> This may be verified to be a continuous extension of {{mvar|f}}.

Equivalently, one can take the [[Stone space]] of the [[complete Boolean algebra]] of all subsets of {{mvar|X}} as the Stone–Čech compactification. This is really the same construction, as the Stone space of this Boolean algebra is the set of ultrafilters (or equivalently prime [[Ideal (order theory)|ideal]]s, or homomorphisms to the 2-element Boolean algebra) of the Boolean algebra, which is the same as the set of ultrafilters on {{mvar|X}}.

The construction can be generalized to arbitrary Tychonoff spaces by using [[Maximal filter|maximal filters]] of [[zero set]]s instead of ultrafilters.<ref>
W.W. Comfort, S. Negrepontis, ''The Theory of Ultrafilters'', Springer, 1974.</ref> (Filters of closed sets suffice if the space is [[normal space|normal]].)

===Construction using C*-algebras===
The Stone–Čech compactification is naturally homeomorphic to the [[Spectrum of a C*-algebra|spectrum]] of C<sub>b</sub>(''X'').<ref>This is Stone's original construction.</ref> Here C<sub>b</sub>(''X'') denotes the [[C*-algebra]] of all continuous bounded [[Complex-valued function|complex-valued functions]] on ''X'' with [[Sup norm|sup-norm]]. Notice that C<sub>b</sub>(''X'') is canonically isomorphic to the [[multiplier algebra]] of C<sub>0</sub>(''X'').

==The Stone–Čech compactification of the natural numbers==
In the case where ''X'' is [[locally compact]], e.g. '''N''' or '''R''', the image of ''X'' forms an open subset of ''βX'', or indeed of any compactification, (this is also a necessary condition, as an open subset of a compact Hausdorff space is locally compact). In this case one often studies the [[Stone–Čech remainder|remainder]] of the space, {{math|''βX'' &setminus; ''X''}}. This is a closed subset of ''βX'', and so is compact. We consider '''N''' with its [[discrete topology]] and write {{math|''β'''''N''' &setminus; '''N''' {{=}} '''N'''*}} (but this does not appear to be standard notation for general ''X'').

As explained above, one can view ''β'''''N''' as the set of [[Ultrafilter (set theory)|ultrafilter]]s on '''N''', with the topology generated by sets of the form <math>\{ F : U \in F \}</math> for ''U'' a subset of '''N'''. The set '''N''' corresponds to the set of [[ultrafilter|principal ultrafilter]]s, and the set '''N'''* to the set of [[ultrafilter|free ultrafilters]].

The study of ''β'''''N''', and in particular '''N'''*, is a major area of modern [[set-theoretic topology]]. The major results motivating this are [[Parovicenko's theorems]], essentially characterising its behaviour under the assumption of the [[continuum hypothesis]].

These state:

* Every compact Hausdorff space of [[weight of a space|weight]] at most <math>\aleph_1</math> (see [[Aleph number]]) is the continuous image of '''N'''* (this does not need the continuum hypothesis, but is less interesting in its absence).
* If the continuum hypothesis holds then '''N'''* is the unique [[Parovicenko space]], up to isomorphism.

These were originally proved by considering [[Boolean algebra (structure)|Boolean algebra]]s and applying [[Stone duality]].

Jan van Mill has described ''β'''''N''' as a "three headed monster"—the three heads being a smiling and friendly head (the behaviour under the assumption of the continuum hypothesis), the ugly head of independence which constantly tries to confuse you (determining what behaviour is possible in different models of set theory), and the third head is the smallest of all (what you can prove about it in [[ZFC]]).<ref>{{Citation| first = Jan| last = van Mill| editor-last = Kunen | editor-first = Kenneth | editor2-last = Vaughan | editor2-first = Jerry E. | contribution = An introduction to βω
 | title = Handbook of Set-Theoretic Topology | year = 1984| pages = 503–560| publisher = North-Holland| isbn = 978-0-444-86580-9}}</ref> It has relatively recently been observed that this characterisation isn't quite right—there is in fact a fourth head of ''β'''''N''', in which [[forcing (mathematics)|forcing axioms]] and Ramsey type axioms give properties of ''β'''''N''' almost diametrically opposed to those under the continuum hypothesis, giving very few maps from '''N'''* indeed. Examples of these axioms include the combination of [[Martin's axiom]] and the [[Open colouring axiom]] which, for example, prove that ('''N'''*)<sup>2</sup> ≠ '''N'''*, while the continuum hypothesis implies the opposite.

=== An application: the dual space of the space of bounded sequences of reals ===
The Stone–Čech compactification ''β'''''N''' can be used to characterize <math>\ell^\infty(\mathbf{N})</math> (the [[Banach space]] of all bounded sequences in the scalar [[Field (mathematics)|field]] '''R''' or '''C''', with [[supremum norm]]) and its [[dual space]].

Given a bounded sequence <math>a\in \ell^\infty(\mathbf{N})</math> there exists a [[closed ball]] ''B'' in the scalar field that contains the image of {{mvar|a}}. {{mvar|a}} is then a function from '''N''' to ''B''. Since '''N''' is discrete and ''B'' is compact and Hausdorff, ''a'' is continuous. According to the universal property, there exists a unique extension ''βa'' : ''β'''''N''' → ''B''. This extension does not depend on the ball ''B'' we consider.

We have defined an extension map from the space of bounded scalar valued sequences to the space of continuous functions over ''β'''''N'''.

:<math> \ell^\infty(\mathbf{N}) \to C(\beta \mathbf{N}) </math>

This map is bijective since every function in ''C''(''β'''''N''') must be bounded and can then be restricted to a bounded scalar sequence.

If we further consider both spaces with the sup norm the extension map becomes an [[isometry]]. Indeed, if in the construction above we take the smallest possible ball ''B'', we see that the sup norm of the extended sequence does not grow (although the image of the extended function can be bigger).

Thus, <math>\ell^\infty(\mathbf{N})</math> can be identified with ''C''(''β'''''N'''). This allows us to use the [[Riesz–Markov–Kakutani_representation_theorem|Riesz representation theorem]] and find that the dual space of <math>\ell^\infty(\mathbf{N})</math> can be identified with the space of finite [[Borel measure]]s on ''β'''''N'''.

Finally, it should be noticed that this technique generalizes to the ''L''<sup>∞</sup> space of an arbitrary [[measure space]] ''X''. However, instead of simply considering the space ''βX'' of ultrafilters on ''X'', the right way to generalize this construction is to consider the [[Stone space]] ''Y'' of the measure algebra of ''X'': the spaces ''C''(''Y'') and ''L''<sup>∞</sup>(''X'') are isomorphic as C*-algebras as long as ''X'' satisfies a reasonable finiteness condition (that any set of positive measure contains a subset of finite positive measure).

===A monoid operation on the Stone–Čech compactification of the naturals===
The [[Natural number|natural numbers]] form a [[monoid]] under [[addition]]. It turns out that this operation can be extended (generally in more than one way, but uniquely under a further condition) to ''β'''''N''', turning this space also into a monoid, though rather surprisingly a non-commutative one.

For any subset, ''A'', of '''N''' and a positive integer ''n'' in '''N''', we define

:<math>A-n=\{k\in\mathbf{N}\mid k+n\in A\}.</math>

Given two ultrafilters ''F'' and ''G'' on '''N''', we define their sum by 

:<math>F+G = \Big\{A\subseteq\mathbf{N}\mid \{n\in\mathbf{N}\mid A-n\in F\}\in G\Big\};</math>

it can be checked that this is again an ultrafilter, and that the operation + is [[associative]] (but not commutative) on β'''N''' and extends the addition on '''N'''; 0 serves as a neutral element for the operation + on ''β'''''N'''. The operation is also right-continuous, in the sense that for every ultrafilter ''F'', the map

:<math>\begin{cases} \beta \mathbf{N} \to \beta \mathbf{N} \\ G \mapsto F+G \end{cases}</math>

is continuous.

More generally, if ''S'' is a [[semigroup]] with the discrete topology, the operation of ''S'' can be extended to ''βS'', getting a right-continuous associative operation.<ref>{{Cite book|last1=Hindman|first1=Neil|title=Algebra in the Stone-Cech Compactification|last2=Strauss|first2=Dona|date=2011-01-21|publisher=DE GRUYTER|isbn=978-3-11-025835-6|location=Berlin, Boston|doi=10.1515/9783110258356}}</ref>

==See also==

* {{annotated link|Compactification (mathematics)}}
* {{annotated link|Filters in topology}}
* {{annotated link|One-point compactification}}
* {{annotated link|Wallman compactification}}

== Notes ==
{{Reflist|group=Note}}

== References ==

* {{citation | last= Čech | first=Eduard | 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}}
* {{cite book|last1=Dunford|first1=Nelson|last2=Schwartz|first2=Jacob T.| title=[[Linear Operators (book)|Linear Operators]], vol. I:general theory|publisher=John Wiley & Sons|date=1988|page=276|edition=Wiley Classics| author1-link=Nelson Dunford |author2-link=Jacob T. Schwarz}}
* {{citation |last1=Hindman |first1=Neil |last2=Strauss |first2=Dona |author-link=Neil Hindman |author2-link=Dona Strauss |title=Algebra in the Stone–Cech compactification. Theory and applications |series=de Gruyter Expositions in Mathematics |volume=27 |publisher=Walter de Gruyter & Co. |location=Berlin |year=1998 |edition=2nd revised and extended 2012 |pages=xiv+485 pp |isbn=978-3-11-015420-7 |mr=1642231 |doi=10.1515/9783110809220}}
* {{Munkres Topology|edition=2}}
* {{springer|title=Stone-Čech compactification|first=I.G. |last=Koshevnikova}}
* {{Narici Beckenstein Topological Vector Spaces|edition=2}} <!-- {{sfn | Narici | 2011 | p=}} --> 
* {{citation | last=Shields | first=Allen | title=Years ago | journal=[[Mathematical Intelligencer]] | volume= 9|issue=2 | year=1987 | pages= 61–63 | doi=10.1007/BF03025901| s2cid=189886579 }}
* {{citation | last= Stone | first=Marshall H. | 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}}
* {{Citation | last1=Tychonoff | first1=Andrey | author-link= Andrey Nikolayevich Tychonoff | title=Über die topologische Erweiterung von Räumen | doi=10.1007/BF01782364 | year=1930 | journal=[[Mathematische Annalen]] | issn=0025-5831 | volume=102 | pages=544–561| s2cid=124737286 }}

==External links==
* ''[http://planetmath.org/stonevcechcompactification Stone-Čech Compactification at Planet Math]''
* Dror Bar-Natan, ''[http://www.math.toronto.edu/~drorbn/classes/9293/131/ultra.pdf Ultrafilters, Compactness, and the Stone–Čech compactification]''

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[[Category:General topology]]
[[Category:Compactification (mathematics)]]