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Stone-cech Compactification
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It has the universal property that any continuous map f : X o Y, where Y is Compact and Hausdorff extends (uniquely) to a continuous map \beta f : \beta X o Y. This universal property may be seen (together with the fact that \beta X is a Hausdorff compactification of X) to characterise \beta X up to isomorphism.

The extension property makes \beta 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 \beta X to Y (for Y in KHauss) correspond bijectively to maps from X to UY (by considering their restriction to X and using the universal property of \beta X). i.e. Hom(\beta X, Y) = Hom(X, UY), which is \beta is left adjoint to U.

One way of constructing \beta X is to consider the map X o 1 ^{C}, where C is the set of continuous functions from X into 1 , given by x o ( f(x) )_{f \in C}. This may be seen to be a homeomorphism onto its image. By Tychonoff's Theorem we have that 1 ^{C} is compact, so the closure of X is a compactification.

In order to verify that this is the Stone-Cech compactification, we just need to verify that it satisfies the appropriate universal property. We do this first for Y = 1 , where the desired extension of f : X o 1 is just the projection onto the f coordinate in [0,1]^C. In order to then get this for general compact Hausdorff Y we use the above to note that Y can be embedded in some cube, extend each of the coordinate functions and then take the product of these extensions.


The Stone-Cech compactification of \mathbb{N}


  • , but this does not appear to be standard for general X.


  • 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} o X with X compact Hausdorff and F an ultrafilter on \mathbb{N} we have an ultrafilter f(F) on X. This has a unique limit, say x, and we define \beta f (F) = x. This may readily be verified to be a continuous extension.

  • 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 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 .

  • 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

  • , while the continuum hypothesis implies the opposite.



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


The Stone-Cëch compactification can be used to caracterize the dual space of l^\infty(\mathbb{N}). Let's consider
\mathbb{N} with the discrete topology and
\beta \mathbb{N} its Stone-Cëch compatification.

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 continuos. According to the universal property, there exists a unique extension
\beta a: \beta \mathbb{N} o B.
This extension does not depend on the ball B we consider.

We have defined an extention map from the space of bounded scalar valued sequences to the space of continuos functions over \beta \mathbb{N} .

l^\infty(\mathbb{N}) o 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 extention 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 tecnique does not apply to the computation of the dual space of L^\infty of an arbitrary measure space X . Although every bounded function can be extended to the Stone-Cëch compactification, not every continuos function will arise in this way since its restriction to X need not to be mesurable. In the case of \mathbb{N} every function (sequence) is mesurable. Futhermore, the space L^\infty(X) is usually taken to be the set of equivalence classes of bounded functions where two of them are said to be equivalent if the differ only in a null-measure set and an undelying measure is implicit. This makes the realization of L^\infty(X) as a spaces of continuos functions even harder if not impossible.