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 space βX. The Stone–Čech compactification βX 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 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. 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 βN \ N is nonempty do not give an explicit description of any particular point in βN \ N.

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

Universal property and functoriality

βX 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 βf: βX → K.

As is usual for universal properties, this universal property, together with the fact that βX is a compact Hausdorff space containing X, characterizes βX up to homeomorphism.

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 the map X → βX need not be a homeomorphism to the image of X (and sometimes is not even injective).

The extension property makes β a functor from Top (the category of topological spaces) to CHaus (the category of compact Hausdorff spaces). 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. Hom(βX, K) = Hom(X, UK), which means that β is left adjoint to U. This implies that CHaus is a reflective subcategory of Top with reflector β.

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

\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 βX is to consider the map

X \to [0, 1]^{C}
x \mapsto ( f(x) )_{f \in C}

where C is the set of all continuous functions from X into [0, 1]. 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 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 βX as the set of all ultrafilters 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 Hausdorff, say x, and we define βf(F) = x. This may be verified to be a continuous extension of f.

Equivalently, one can take the Stone space of the complete 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 sets instead of ultrafilters. (Filters of closed sets suffice if the space is normal.)

Construction using C*-algebras

The Stone–Čech compactification is naturally homeomorphic to the spectrum of Cb(X).[1] Here Cb(X) denotes the C*-algebra of all continuous bounded functions on X with sup-norm. Notice that Cb(X) is canonically isomorphic to the multiplier algebra of C0(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 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).

One can view βN as the set of ultrafilters on N, with the topology generated by sets of the form \{ F : U \in F \} for U a subset of N. The set N corresponds to the set of principal ultrafilters, and the set N* to the set of free ultrafilters.

The easiest way to see this is isomorphic to βN is to show that it satisfies the universal property. For f: N → K with K compact Hausdorff and F an ultrafilter on 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 βf(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 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 state:

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).[2] 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*)2 ≠ 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 βN can be used to characterize ℓ∞(N) (the Banach space of all bounded sequences in the scalar field R or C, with supremum norm) and its dual space.

Given a bounded sequence a in ℓ∞(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 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: βN → 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 βN.

\ell^\infty(\mathbf{N}) \to C(\beta \mathbf{N})

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).

Thus, ℓ∞(N) can be identified with C(βN). This allows us to use the Riesz representation theorem and find that the dual space of ℓ∞(N) can be identified with the space of finite Borel measures on βN.

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).

Addition on the Stone–Čech compactification of the naturals

The natural numbers form a monoid under addition. It turns out that this operation can be extended (in more than one way) to βN, 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

A-n=\{k\in\mathbf{N}\mid k+n\in A\}.

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

F+G = \Big\{A\subset\mathbf{N}\mid \{n\in\mathbf{N}\mid A-n\in F\}\in G\Big\};

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

\beta \mathbf{N}\to\beta \mathbf{N}
G \mapsto F+G

is continuous.

See also

Notes

  1. ↑ This is Stone's original construction.
  2. ↑ van Mill, Jan (1984), "An introduction to βω", in Kunen, Kenneth; Vaughan, Jerry E., Handbook of Set-Theoretic Topology, North-Holland, pp. 503–560, ISBN 0-444-86580-2

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