- 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 a compact Hausdorff spaceβ"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 a Tychonoff spacethen 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 choiceis required to prove that every topological space has a Stone–Čech compactification. Even for quite simple spaces , an accessible concrete description of often remains elusive. In particular it is impossible to explicitly exhibit a point in .
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: any continuous 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 a
functorfrom Top (the category of topological spaces) to CHaus (the category of compact Hausdorff spaces). If we let be the inclusion functorfrom CHaus into Top, maps from to (for in CHaus) correspond bijectively to maps from to (by considering their restriction to and using the universal property of ). i.e. , which means that is left adjoint to . This implies that CHaus is a reflective subcategoryof Top with reflector .
Construction using products
One attempt to construct the Stone-Cech compactification of "X" is to take the closure of the image of "X" in: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 is to consider the
:where is the set of all
continuous functions from into . This may be seen to be a continuous map onto its image, if is given the product topology. By Tychonoff's theoremwe have that is compact since [0,1] is, so the closure of in is a compactification of .
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 , where the desired extension of "f" : "X" → [0,1] is just the projection onto the coordinate in . In order to then get this for general compact Hausdorff we use the above to note that 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 is discrete, one can construct as the set of all
ultrafilters on , with a topology known as "Stone topology". The elements of correspond to the principal ultrafilters.
Again we verify the universal property: For "f" : "X" → "K" with compact Hausdorff and an ultrafilter on we have an ultrafilter on . This has a unique limit because is compact, say , and we define . This may be verified to be a continuous extension of .
Equivalently, one can take the
Stone spaceof the complete Boolean algebraof 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 sets 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(Cb("X")). Here Cb("X") denotes the
C*-algebraof all continuous bounded functions on "X" with sup-norm, and M(Cb("X")) denotes its multiplier algebra.
The Stone–Čech compactification of the natural numbers
In the case where is
locally compact, e.g. or , it forms an open subset of , 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, . This is a closed subset of , and so is compact. We consider with its discrete topologyand write (but this does not appear to be standard notation for general ).
The easiest way to see this is isomorphic to is to show that it satisfies the universal property. For with compact Hausdorff and an ultrafilter on we have an ultrafilter on , the pushforward of . This has a unique limit, say , because is compact Hausdorff, and we define . 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 .)
The study of , and in particular , 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.
* Every compact Hausdorff space of weight at most (see
Aleph number) is the continuous image of (this does not need the continuum hypothesis, but is less interesting in its absence).
* If the continuum hypothesis holds then is the unique
Parovicenko space, up to isomorphism.
These were originally proved by considering Boolean algebras and applying
Jan van Mill has described 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 , in which forcing axioms and Ramsey type axioms give properties of almost diametrically opposed to those under the continuum hypothesis, giving very few maps from indeed. Examples of these axioms include the combination of Martin's axiomand the Open colouring axiomwhich, for example, prove that , while the continuum hypothesis implies the opposite.
An application: the dual space of the space of bounded sequences of reals
The Stone–Čech compactification can be used to characterize (the
Banach spaceof all bounded sequences in the scalar field R or C, with supremum norm) and its dual space.
Given a bounded sequence , there exists a closed ball that contains the image of ( is a subset of the scalar field). is then a function from to . Since is discrete and is compact and Hausdorff, is continuous. According to the universal property, there exists a unique extension.This extension does not depend on the ball we consider.
We have defined an extension map from the space of bounded scalar valued sequences to the space of continuous functions over .
This map is bijective since every function in 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 , we see that the sup norm of the extended sequence does not grow (although the image of the extended function can be bigger).
Thus, can be identified with. This allows us to use the
Riesz representation theoremand find that the dual space of can be identified with the space of finite Borel measures on .
Finally, it should be noticed that this technique generalizes to the space of an arbitrary
measure space. However, instead of simply considering the space of ultrafilters on , the right way to generalize this construction is to consider the Stone spaceof the measure algebra of : the spaces and are isomorphic as C*-algebras as long as 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
monoidunder addition. It turns out that this operation can be extended (in more than one way) to , turning this space also into a monoid, though rather surprisingly a non-commutative one.
For any subset and , we define:Given two ultrafilters and on , we define their sum by :it can be checked that this is again an ultrafilter, and that the operation + is
associative(but not commutative) on and extends the addition on ; 0 serves as a neutral element for the operation + on . The operation is also right-continuous, in the sense that for every ultrafilter , the
* Dror Bar-Natan, " [http://www.math.toronto.edu/~drorbn/classes/9293/131/ultra.pdf Ultrafilters, Compactness, and the Stone–Čech compactification] "
*citation|first=E.|last= Čech|title=On bicompact spaces|journal= Ann. of Math. |volume= 38 |year=1937 |pages= 823–844
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
*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
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