Stone–Čech compactification

In the mathematical discipline of general topology, Stone–Čech compactification (or Čech–Stone compactification[1]) is a technique for constructing a universal map from a topological space X to a compact Hausdorff space βX. 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 factors through βX (in a unique way). If X is a Tychonoff space then the map from X to its image in βX is a homeomorphism, so X can be thought of as a (dense) subspace of βX; every other compact Hausdorff space that densely contains X is a quotient of βX. For general topological spaces X, the map from X to βX 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 βX often remains elusive. In particular, proofs that βX \ X is nonempty do not give an explicit description of any particular point in βX \ X.

The Stone–Čech compactification occurs implicitly in a paper by Andrey Nikolayevich Tychonoff (1930) and was given explicitly by Marshall Stone (1937) and Eduard Čech (1937).

Andrey Nikolayevich Tikhonov introduced completely regular spaces in 1930 in order to avoid the pathological situation of Hausdorff spaces whose only continuous real-valued functions are constant maps.[2]

In the same 1930 article where Tychonoff defined completely regular spaces, he also proved that every Tychonoff space (i.e. Hausdorff completely regular space) has a Hausdorff 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.[3]

The Stone–Čech compactification of the topological space X is a compact Hausdorff space βX together with a continuous map iX : XβX that has the following universal property: any continuous map f : XK, where K is a compact Hausdorff space, extends uniquely to a continuous map βf : βXK, i.e. (βf)iX = f .

As is usual for universal properties, this universal property characterizes βX up to homeomorphism.

As is outlined in § Constructions, below, one can prove (using the axiom of choice) that such a Stone–Čech compactification iX : XβX exists for every topological space X. Furthermore, the image iX(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 Stone–Čech construction can be performed for more general spaces X, but in that case the map XβX 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 bijectively to maps from X to UK (by considering their restriction to X and using the universal property of βX). i.e.

which means that β is left adjoint to U. This implies that CHaus is a reflective subcategory of Top with reflector β.

If X is a compact Hausdorff space, then it coincides with its Stone–Čech compactification. Most other Stone–Čech compactifications lack concrete descriptions and are extremely unwieldy.[citation needed] Exceptions include:

One attempt to construct the Stone–Čech 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 K. 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 set to which X can be mapped with dense image.

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

The construction can be generalized to arbitrary Tychonoff spaces by using maximal filters of zero sets instead of ultrafilters.[5] (Filters of closed sets suffice if the space is normal.)

The Stone–Čech compactification is naturally homeomorphic to the spectrum of Cb(X).[6] Here Cb(X) denotes the C*-algebra of all continuous bounded complex-valued functions on X with sup-norm. Notice that Cb(X) is canonically isomorphic to the multiplier algebra of C0(X).

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 remainder of the space, βX \ X. This is a closed subset of βX, and so is compact. We consider N with its discrete topology and write βN \ N = N* (but this does not appear to be standard notation for general X).

The study of βN, 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 were originally proved by considering Boolean algebras and applying Stone duality.

Jan van Mill has described βN 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).[7] It has relatively recently been observed that this characterisation isn't quite right—there is in fact a fourth head of βN, in which forcing axioms and Ramsey type axioms give properties of βN 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*)2N*, while the continuum hypothesis implies the opposite.

An application: the dual space of the space of bounded sequences of reals

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

This map is bijective since every function in C(βN) 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).

Finally, it should be noticed that this technique generalizes to the L 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(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).

The 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 βN, turning this space also into a monoid, though rather surprisingly a non-commutative one.

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

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.[8]