- Stone–Čech 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 acompact Hausdorff space β"X". The Stone–Čech compactification β"X" of a topological space "X" is the largest compact Hausdorff space "generated" by "X", in the sense that any map from "X" to a compact Hausdorff space factors through β"X" (in a unique way). If "X" is aTychonoff space then the map from "X" to its image in β"X" is a homeomorphism, so "X" can be thought of as a (dense) subspace of β"X". For general topological spaces "X", the map from "X" to β"X" need not be 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 $eta\; X$ often remains elusive. In particular it is impossible to explicitly exhibit a point in $eta\; mathbb\{N\}\; setminus\; mathbb\{N\}$.The Stone–Čech compactification was found 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.

**Universal property and functoriality**β"X" is a compact Hausdorff space together with a continuous map from "X" and has the following

universal property : anycontinuous map "f" : "X" → "K", where "K" is a compact Hausdorff space, lifts uniquely to a continuous map β"f" : β"X" → "K". :As is usual for universal properties, this universal property, together with the fact that β"X" is a compact Hausdorff space containing "X", characterizes β"X"up to homeomorphism .Some authors add the assumption that the starting space be Tychonoff (or even locally compact Hausdorff), for the following reasons:

*The map from "X" to its image in β"X" is a homeomorphism if and only if "X" is Tychonoff.

*The map from "X" to its image in β"X" 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 the map "X" → β"X" need not be a homeomorphism to the image of "X" (and sometimes is not even injective).The extension property makes $eta$ a

functor from**Top**(the category of topological spaces) to**CHaus**(the category of compact Hausdorff spaces). If we let $U$ be theinclusion functor from**CHaus**into**Top**, maps from $eta\; X$ to $K$ (for $K$ in**CHaus**) correspond bijectively to maps from $X$ to $UK$ (by considering their restriction to $X$ and using the universal property of $eta\; X$). i.e. $mbox\{Hom\}(eta\; X,\; K)\; =\; mbox\{Hom\}(X,\; UK)$, which means that $eta$ is left adjoint to $U$. This implies that**CHaus**is areflective subcategory of**Top**with reflector $eta$.**Constructions****Construction using products**One attempt to construct the Stone-Cech compactification of "X" is to take the closure of the image of "X" in:$prod\; C$where the product is over all maps from "X" to compact Hausdorff spaces "C". 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 "C" 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 set to which "X" can be mapped with dense image.

**Construction using the unit interval**One way of constructing $eta\; X$ is to consider the

$X\; o\; [0,\; 1]\; ^\{C\}$ :$x\; mapsto\; (\; f(x)\; )\_\{f\; in\; C\}$where $C$ is the set of allcontinuous function s from $X$ into $[0,\; 1]$. This may be seen to be a continuous map onto its image, if $[0,\; 1]\; ^\{C\}$ is given theproduct topology . ByTychonoff's theorem we have that $[0,\; 1]\; ^\{C\}$ is compact since [0,1] is, so the closure of $X$ in $[0,\; 1]\; ^\{C\}$ is a compactification of $X$.In order to verify that this is the Stone–Čech compactification, we just need to verify that it 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 $[0,1]\; ^C$. 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 "f" and "g" are any two distinct maps from compact Hausdorff spaces "A" to "B", then there is a map "h" from "B" to [0,1] such that "hf" and "hg" are distinct. Any other cogenerator (or cogenerating set) can be used in this construction.**Construction using ultrafilters**Alternatively, if $X$ is discrete, one can construct $eta\; X$ as the set of all

ultrafilter s on $X$, with a topology known as "Stone topology". The elements of $X$ correspond to the principal ultrafilters.Again we verify the universal property: For "f" : "X" → "K" with $K$ compact Hausdorff and $F$ an ultrafilter on $X$ we have an ultrafilter $f(F)$ on $K$. This has a unique limit because $K$ is compact, say $x$, and we define $eta\; f\; (F)\; =\; x$. This may be verified to be a continuous extension of $f$.

Equivalently, one can take the

Stone space of thecomplete Boolean algebra of all subsets of "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 ideals, or homomorphisms to the 2 element Boolean algebra) of the Boolean algebra, which is the same as the set of ultrafilters on "X".The construction can be generalized to arbitrary Tychonoff spaces by using maximal filters of

zero set s instead of ultrafilters. (Filters of closed sets suffice if the space is normal.)**Construction using C*-algebras**In case "X" is a completely regular Hausdorff space, the Stone-Čech compactification can be identified with the spectrum of M(C

_{b}("X")). Here C_{b}("X") denotes theC*-algebra of all continuous bounded functions on "X" with sup-norm, and M(C_{b}("X")) denotes itsmultiplier algebra .**The Stone–Čech compactification of the natural numbers**In the case where $X$ is

locally compact , e.g. $mathbb\{N\}$ or $mathbb\{R\}$, it forms an open subset of $eta\; X$, 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 remainder of the space, $eta\; X\; setminus\; X$. This is a closed subset of $eta\; X$, and so is compact. We consider $mathbb\{N\}$ with itsdiscrete topology and write $eta\; mathbb\{N\}\; setminus\; mathbb\{N\}\; =\; mathbb\{N\}^*$ (but this does not appear to be standard notation for general $X$).One can view $eta\; mathbb\{N\}$ as the set of

ultrafilter s on $mathbb\{N\}$, with the topology generated by sets of the form $\{\; F\; :\; U\; in\; F\; \}$ for $U\; subseteq\; mathbb\{N\}$. $mathbb\{N\}$ corresponds to the set of principal ultrafilters, and $mathbb\{N\}^*$ to the set of free ultrafilters.The easiest way to see this is isomorphic to $eta\; mathbb\{N\}$ is to show that it satisfies the universal property. For $f\; :\; mathbb\{N\}\; o\; K$ with $K$ compact Hausdorff and $F$ an ultrafilter on $mathbb\{N\}$ we have an ultrafilter $f(F)$ on $K$, the pushforward of $F$. This has a unique limit, say $x$, because $K$ is compact Hausdorff, and we define $eta\; f\; (F)\; =\; x$. This may readily be verified to be a continuous extension.

(A similar but slightly more involved construction of the Stone–Čech compactification as a set of certain maximal filters can also be given for a general Tychonoff space $X$.)

The study of $eta\; N$, and in particular $mathbb\{N\}^*$, 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 thecontinuum hypothesis .These state:

* Every compact Hausdorff space of weight at most $aleph\_1$ (see

Aleph number ) is the continuous image of $mathbb\{N\}^*$ (this does not need the continuum hypothesis, but is less interesting in its absence).

* If the continuum hypothesis holds then $mathbb\{N\}^*$ is the uniqueParovicenko space , up to isomorphism.These were originally proved by considering Boolean algebras and applying

Stone duality .Jan van Mill has described $eta\; mathbb\{N\}$ 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 ). It has relatively recently been observed that this characterisation isn't quite right - there is in fact a fourth head of $eta\; mathbb\{N\}$, in which forcing axioms and Ramsey type axioms give properties of $eta\; mathbb\{N\}$ almost diametrically opposed to those under the continuum hypothesis, giving very few maps from $mathbb\{N\}^*$ indeed. Examples of these axioms include the combination ofMartin's axiom and theOpen colouring axiom which, for example, prove that $(mathbb\{N\}^*)^2\; ot=\; mathbb\{N\}^*$, while the continuum hypothesis implies the opposite.**An application: the dual space of the space of bounded sequences of reals**The Stone–Čech compactification $eta\; N$ can be used to characterize $l^infty(mathbb\{N\})$ (the

Banach space of all bounded sequences in the scalar field**R**or**C**, withsupremum norm ) and itsdual space .Given a bounded sequence $a\; in\; l^infty(mathbb\{N\})$, there exists a closed ball $B$ that contains the image of $a$ ($B$ is a subset of the scalar field).$a$ is then a function from $mathbb\{N\}$ to $B$. Since $mathbb\{N\}$ is discrete and$B$ is compact and Hausdorff,$a$ is continuous. According to the universal property, there exists a unique extension$eta\; a:\; eta\; mathbb\{N\}\; o\; B$.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 $eta\; mathbb\{N\}$.

:$l^infty(mathbb\{N\})\; o\; C(eta\; mathbb\{N\})$

This map is bijective since every function in$C(eta\; mathbb\{N\})$ 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, $l^infty(mathbb\{N\})$ can be identified with$C(eta\; mathbb\{N\})$. This allows us to use the

Riesz representation theorem and find that the dual space of $l^infty(mathbb\{N\})$ can be identified with the space of finiteBorel measure s on $eta\; mathbb\{N\}$.Finally, it should be noticed that this technique generalizes to the $L^infty$ space of an arbitrary

measure space $X$. However, instead of simply considering the space $eta\; X$ of ultrafilters on $X$, the right way to generalize this construction is to consider theStone space $Y$ of the measure algebra of $X$: the spaces $C(Y)$ and $L^infty(X)$ 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).**Addition on the Stone–Čech compactification of the naturals**The natural numbers form a

monoid underaddition . It turns out that this operation can be extended (in more than one way) to $eta\; mathbb\{N\}$, turning this space also into a monoid, though rather surprisingly a non-commutative one.For any subset $A\; subset\; mathbb\{N\}$ and $ninmathbb\{N\}$, we define:$A-n=\{kinmathbb\{N\}mid\; k+nin\; A\}.$Given two ultrafilters $F$ and $G$ on $mathbb\{N\}$, we define their sum by :$F+G\; =\; Big\{Asubsetmathbb\{N\}mid\; \{ninmathbb\{N\}mid\; A-nin\; F\}in\; GBig\};$it can be checked that this is again an ultrafilter, and that the operation + is

associative (but not commutative) on $eta\; mathbb\{N\}$ and extends the addition on $mathbb\{N\}$; 0 serves as a neutral element for the operation + on $eta\; mathbb\{N\}$. The operation is also right-continuous, in the sense that for every ultrafilter $F$, the

$eta\; mathbb\{N\}\; oeta\; mathbb\{N\}$:$G\; mapsto\; F+G$is continuous.**ee also***

One-point compactification

*Wallman compactification **External links***

* Dror Bar-Natan, " [*http://www.math.toronto.edu/~drorbn/classes/9293/131/ultra.pdf Ultrafilters, Compactness, and the Stone–Čech compactification*] "**References***citation|first=E.|last= Čech|title=On bicompact spaces|journal= Ann. of Math. |volume= 38 |year=1937 |pages= 823–844

url=http://links.jstor.org/sici?sici=0003-486X%28193710%292%3A38%3A4%3C823%3AOBS%3E2.0.CO%3B2-P

*citation|id=MR|1642231

last=Hindman|first= Neil|last2= Strauss|first2= Dona

title=Algebra in the Stone-Cech compactification. Theory and applications |series=de Gruyter Expositions in Mathematics|volume= 27|publisher= Walter de Gruyter & Co.|publication-place= Berlin|year= 1998|pages= xiv+485 pp. |ISBN= 3-11-015420-X

*springer|id=S/s090340|first=I.G. |last=Koshevnikova

*citation|first=M.H.|last= Stone|title=Applications of the theory of Boolean rings to general topology |journal=Trans. Amer. Soc. |volume= 41 |year=1937|pages= 375–481

url=http://links.jstor.org/sici?sici=0002-9947%28193705%2941%3A3%3C375%3AAOTTOB%3E2.0.CO%3B2-8

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