Editing Stone–Čech compactification (section)

==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 [[Stone–Čech remainder|remainder]] of the space, {{math|''βX'' &setminus; ''X''}}. This is a closed subset of ''βX'', and so is compact. We consider '''N''' with its [[discrete topology]] and write {{math|''β'''''N''' &setminus; '''N''' {{=}} '''N'''*}} (but this does not appear to be standard notation for general ''X'').

As explained above, one can view ''β'''''N''' as the set of [[Ultrafilter (set theory)|ultrafilter]]s on '''N''', with the topology generated by sets of the form <math>\{ F : U \in F \}</math> for ''U'' a subset of '''N'''. The set '''N''' corresponds to the set of [[ultrafilter|principal ultrafilter]]s, and the set '''N'''* to the set of [[ultrafilter|free ultrafilters]].

The study of ''β'''''N''', 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:

* Every compact Hausdorff space of [[weight of a space|weight]] at most <math>\aleph_1</math> (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 algebra (structure)|Boolean algebra]]s and applying [[Stone duality]].

Jan van Mill has described ''β'''''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]]).<ref>{{Citation| first = Jan| last = van Mill| editor-last = Kunen | editor-first = Kenneth | editor2-last = Vaughan | editor2-first = Jerry E. | contribution = An introduction to βω
 | title = Handbook of Set-Theoretic Topology | year = 1984| pages = 503–560| publisher = North-Holland| isbn = 978-0-444-86580-9}}</ref> It has relatively recently been observed that this characterisation isn't quite right—there is in fact a fourth head of ''β'''''N''', in which [[forcing (mathematics)|forcing axioms]] and Ramsey type axioms give properties of ''β'''''N''' 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'''*)<sup>2</sup> ≠ '''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 ''β'''''N''' can be used to characterize <math>\ell^\infty(\mathbf{N})</math> (the [[Banach space]] of all bounded sequences in the scalar [[Field (mathematics)|field]] '''R''' or '''C''', with [[supremum norm]]) and its [[dual space]].

Given a bounded sequence <math>a\in \ell^\infty(\mathbf{N})</math> there exists a [[closed ball]] ''B'' in the scalar field that contains the image of {{mvar|a}}. {{mvar|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'' : ''β'''''N''' → ''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 ''β'''''N'''.

:<math> \ell^\infty(\mathbf{N}) \to C(\beta \mathbf{N}) </math>

This map is bijective since every function in ''C''(''β'''''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 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, <math>\ell^\infty(\mathbf{N})</math> can be identified with ''C''(''β'''''N'''). This allows us to use the [[Riesz–Markov–Kakutani_representation_theorem|Riesz representation theorem]] and find that the dual space of <math>\ell^\infty(\mathbf{N})</math> can be identified with the space of finite [[Borel measure]]s on ''β'''''N'''.

Finally, it should be noticed that this technique generalizes to the ''L''<sup>∞</sup> 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''<sup>∞</sup>(''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 number|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 ''β'''''N''', 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

:<math>A-n=\{k\in\mathbf{N}\mid k+n\in A\}.</math>

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

:<math>F+G = \Big\{A\subseteq\mathbf{N}\mid \{n\in\mathbf{N}\mid A-n\in F\}\in G\Big\};</math>

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

:<math>\begin{cases} \beta \mathbf{N} \to \beta \mathbf{N} \\ G \mapsto F+G \end{cases}</math>

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.<ref>{{Cite book|last1=Hindman|first1=Neil|title=Algebra in the Stone-Cech Compactification|last2=Strauss|first2=Dona|date=2011-01-21|publisher=DE GRUYTER|isbn=978-3-11-025835-6|location=Berlin, Boston|doi=10.1515/9783110258356}}</ref>