Stone–Čech compactification

In the mathematical discipline of

.

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

).

History

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]

Universal property and functoriality

The Stone–Čech compactification of the topological space X is a compact Hausdorff space βX together with a continuous map i_X : X → βX that has the following

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 i_X : X → βX exists for every topological space X. Furthermore, the image i_X(X) is dense in βX.

Some authors add the assumption that the starting space X be Tychonoff (or even

Hausdorff), for the following reasons:

• The map from X to its image in βX is a homeomorphism if and only if X is Tychonoff.
• The map from X to its image in βX 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 in that case the map X → βX 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

to X and using the universal property of βX). i.e.

Hom(βX, K) ≅ Hom(X, UK),

which means that β is

of Top with reflector β.

Examples

If X is a compact Hausdorff space, then it coincides with its Stone–Čech compactification.[5]

The Stone–Čech compactification of the first uncountable ordinal ${\displaystyle \omega _{1}}$, with the order topology, is the ordinal ${\displaystyle \omega _{1}+1}$. The Stone–Čech compactification of the

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

${\displaystyle \prod \nolimits _{f:X\to K}K}$

where the product is over all maps from X to compact Hausdorff spaces K (or, equivalently, the image of X by the right

of the power set of X), which is sufficiently large that it has cardinality at least equal to that of every compact Hausdorff space to which X can be mapped with dense image.

Construction using the unit interval

One way of constructing βX is to let C be the set of all continuous functions from X into [0, 1] and consider the map ${\displaystyle e:X\to [0,1]^{C}}$ where

${\displaystyle e(x):f\mapsto f(x)}$

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.

Construction using ultrafilters

Alternatively, if ${\displaystyle X}$ is discrete, then it is possible to construct ${\displaystyle \beta X}$ as the set of all

on ${\displaystyle X,}$ with the elements of ${\displaystyle X}$ corresponding to the , is generated by sets of the form ${\displaystyle \{F:U\in F\}}$ for ${\displaystyle U}$ a subset of ${\displaystyle X.}$

Again we verify the universal property: For ${\displaystyle f:X\to K}$ with ${\displaystyle K}$ compact Hausdorff and ${\displaystyle F}$ an ultrafilter on ${\displaystyle X}$ we have an ultrafilter base ${\displaystyle f(F)}$ on ${\displaystyle K,}$ the pushforward of ${\displaystyle F.}$ This has a unique limit because ${\displaystyle K}$ is compact Hausdorff, say ${\displaystyle x,}$ and we define ${\displaystyle \beta f(F)=x.}$ This may be verified to be a continuous extension of ${\displaystyle f.}$

Equivalently, one can take the Stone space of the complete Boolean algebra of all subsets of ${\displaystyle X}$ 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 ${\displaystyle X.}$

The construction can be generalized to arbitrary Tychonoff spaces by using

.)

Construction using C*-algebras

The Stone–Čech compactification is naturally homeomorphic to the

of C₀(X).

The Stone–Čech compactification of the natural numbers

In the case where X is

discrete topology

and write βN \ N = N* (but this does not appear to be standard notation for general X).

As explained above, one can view βN as the set of

ultrafilters

on N, with the topology generated by sets of the form ${\displaystyle \{F:U\in F\}}$ 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 study of βN, and in particular N*, is a major area of modern

.

These state:

• Every compact Hausdorff space of
  weight
  at most ${\displaystyle \aleph _{1}}$ (see Aleph number) is the continuous image of N* (this does not need the continuum hypothesis, but is less interesting in its absence).
• If the continuum hypothesis holds then N* is the unique Parovicenko space, up to isomorphism.

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

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 ${\displaystyle \ell ^{\infty }(\mathbf {N} )}$ (the

.

Given a bounded sequence ${\displaystyle a\in \ell ^{\infty }(\mathbf {N} )}$ there exists a

closed ball

B in the scalar field that contains the image of ${\displaystyle a}$. ${\displaystyle a}$ 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.

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

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, ${\displaystyle \ell ^{\infty }(\mathbf {N} )}$ can be identified with C(βN). This allows us to use the Riesz representation theorem and find that the dual space of ${\displaystyle \ell ^{\infty }(\mathbf {N} )}$ 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).

A monoid operation 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 (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.

For any subset, A, of N and a positive integer n in N, we define

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

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

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

it can be checked that this is again an ultrafilter, and that the operation + is

(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

${\displaystyle {\begin{cases}\beta \mathbf {N} \to \beta \mathbf {N} \\G\mapsto F+G\end{cases}}}$

is continuous.

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.^[10]

See also

• Compactification (mathematics) – Embedding a topological space into a compact space as a dense subset
• Filters in topology – Use of filters to describe and characterize all basic topological notions and results.
• One-point compactification
   – Way to extend a non-compact topological spacePages displaying short descriptions of redirect targets
• Wallman compactification – A compactification of T₁ topological spaces

Notes

References

• JSTOR 1968839
• Schwarz, Jacob T.
  (1988). Linear Operators part I:general theory (Wiley Classics ed.). John Wiley & Sons. p. 276.
• MR 1642231
• OCLC 42683260
  .
• Koshevnikova, I.G. (2001) [1994], "Stone-Čech compactification", Encyclopedia of Mathematics, EMS Press
• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press.
  OCLC 144216834
  .
• Shields, Allen (1987), "Years ago",
  S2CID 189886579
• Stone, Marshall H. (1937), "Applications of the theory of Boolean rings to general topology",
  JSTOR 1989788
• S2CID 124737286

External links

• Stone-Čech Compactification at Planet Math
• Dror Bar-Natan, Ultrafilters, Compactness, and the Stone–Čech compactification