Stone–Čech compactification

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In mathematics, Stone–Čech compactification is a technique for embedding a Tychonoff topological space into a compact Hausdorff space. The Stone–Čech compactification $β X$ of a Tychonoff space $X$ is the largest Hausdorff compactification of $X$ , in the sense that any Hausdorff compactification of $X$ is a quotient of $β X$ in a way that preserves the embeddings of $X$ .

A form of the axiom of choice is required to prove that every Tychonoff space has a Stone–Čech compactification. Even for quite simple spaces $X$ , an accessible concrete description of $β X$ often remains elusive. In particular it is impossible to explicitly exhibit a point in $\beta \mathbb{N} \setminus \mathbb{N}$ .

1 Universal property and functoriality
2 Construction
3 The Stone–Čech compactification of
- 3.1 An application: the dual space of
- 3.2 Addition on
4 History
5 References

[edit] Universal property and functoriality

$β X$ is a compact Hausdorff space that contains $X$ as a dense subspace and 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. 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.

The extension property makes $β$ a functor from Tych (the category of Tychonoff spaces) into KHauss (the category of compact Hausdorff spaces). If we let $U$ be the inclusion functor from KHauss into Tych, maps from $β X$ to $K$ (for $K$ in KHauss) correspond bijectively to maps from $X$ to $U K$ (by considering their restriction to $X$ and using the universal property of $β X$ ). i.e. $H o m (β X, Y) = H o m (X, U Y)$ , which means that $β$ is left adjoint to $U$ .

Note that the assumption that the starting space be Tychonoff is natural, because one can show that a space $X$ can be embedded into a compact Hausdorff space if and only if $X$ is Tychonoff. The Stone–Čech construction can be performed for more general spaces $X$ , but the map X → βX is then not injective anymore.

[edit] Construction

One way of constructing $β X$ 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 homeomorphism 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, 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.

Alternatively, if $X$ is discrete, one can construct $β X$ 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, say $x$ , and we define $β f (F) = x$ . This may be verified to be a continuous extension of $f$ .

In case X is a locally compact space, the Stone-Čech compactification can be identified with the spectrum of M(C₀(X)). Here C₀(X) denotes the C*-algebra of all continuous functions on X which vanish at infinity, and M(C₀(X)) denotes its multiplier algebra.

[edit] The Stone–Čech compactification of $\mathbb{N}$

In the case where $X$ is locally compact, e.g. $\mathbb{N}$ or $\mathbb{R}$ , it forms an open subset of $β X$ , 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, $\beta X \setminus X$ . This is a closed subset of $β X$ , and so is compact. We consider $\mathbb{N}$ with its discrete topology and write $\beta \mathbb{N} \setminus \mathbb{N} = \mathbb{N}^*$ (but this does not appear to be standard notation for general $X$ ).

One can view $\beta \mathbb{N}$ as the set of ultrafilters 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 $\beta \mathbb{N}$ is to show that it satisfies the universal property. For $f : \mathbb{N} \to K$ 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 $β f (F) = x$ . 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 $β N$ , 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 the continuum 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 unique Parovicenko space, up to isomorphism.

These were originally proved by using Boolean algebra methods and applying Stone duality.

Jan van Mill has described $\beta \mathbb{N}$ 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 $\beta \mathbb{N}$ , in which forcing axioms and Ramsey type axioms give properties of $\beta \mathbb{N}$ almost diametrically opposed to those under the continuum hypothesis, giving very few maps from $\mathbb{N}^*$ indeed. Examples of these axioms include the combination of Martin's axiom and the Open colouring axiom which, for example, prove that $(\mathbb{N}^*)^2 \not= \mathbb{N}^*$ , while the continuum hypothesis implies the opposite.

[edit] An application: the dual space of $l^\infty(\mathbb{N})$

The Stone–Čech compactification $β N$ can be used to characterize $l^\infty(\mathbb{N})$ (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 $a \in l^\infty(\mathbb{N})$ , 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 $\beta a: \beta \mathbb{N} \to B$ . 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 $\beta \mathbb{N}$ .

$l^\infty(\mathbb{N}) \to C(\beta \mathbb{N})$

This map is bijective since every function in $C(\beta \mathbb{N})$ 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 becames 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 $C(\beta \mathbb{N})$ . 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 finite Borel measures on $\beta \mathbb{N}$ .

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 $β X$ 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 $L^\infty(X)$ 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).

[edit] Addition on $\beta \mathbb{N}$

The natural numbers form a monoid under addition. It turns out that this operation can be extended to $\beta \mathbb{N}$ , turning this space also into a monoid.

For any subset $A \subset \mathbb{N}$ and $n\in\mathbb{N}$ , we define

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

Given two ultrafilters $F$ and $G$ on $\mathbb{N}$ , we define their sum by

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

it can be checked that this is again an ultrafilter, and that the operation + is associative on $\beta \mathbb{N}$ and extends the addition on $\mathbb{N}$ ; 0 serves as a neutral element for the operation + on $\beta \mathbb{N}$ . The operation is also right-continuous, in the sense that for every ultrafilter $F$ , the map

$\beta \mathbb{N}\to\beta \mathbb{N}$

$G \mapsto F+G$

is continuous.

[edit] History

The Stone–Čech compactification is named after Marshall Stone, who wrote about it in his article Applications of the theory of Boolean rings to general topology (1937), and Eduard Čech, who wrote On bicompact spaces also in 1937.