Stone–Čech compactification
In the mathematical discipline of general topology, Stone–Čech compactification (or Čech–Stone compactification[1]) is a technique for constructing a universal map from a topological space X to a compact Hausdorff space 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.
The Stone–Čech compactification occurs implicitly in a paper by Andrey Nikolayevich Tychonoff (1930) and was given explicitly by Marshall Stone (1937) and Eduard Čech (1937).
Andrey Nikolayevich Tikhonov introduced completely regular spaces in 1930 in order to avoid the pathological situation of Hausdorff spaces whose only continuous real-valued functions are constant maps.
In 1937, Čech extended Tychonoff's technique and introduced the notation βX for this compactification.
[3] The Stone–Čech compactification of the topological space X is a compact Hausdorff space βX together with a continuous map iX : X → βX that 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, i.e. (βf)iX = f.[4] 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 iX : X → βX exists for every topological space X.
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 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 functor from Top (the category of topological spaces) to CHaus (the category of compact Hausdorff spaces).
Further, 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).
This implies that CHaus is a reflective subcategory of Top with reflector β.
[6] One attempt to construct the Stone–Čech compactification of X is to take the closure of the image of X in where the product is over all maps from X to compact Hausdorff spaces K (or, equivalently, the image of X by the right Kan extension of the identity functor of the category CHaus of compact Hausdorff spaces along the inclusion functor of CHaus into the category Top of general topological spaces).
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 K 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 space to which X can be mapped with dense image.
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
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.
Equivalently, one can take the Stone space of the complete Boolean algebra of all subsets of
The construction can be generalized to arbitrary Tychonoff spaces by using maximal filters of zero sets instead of ultrafilters.
[8] Here Cb(X) denotes the C*-algebra of all continuous bounded complex-valued functions on X with sup-norm.
We consider N with its discrete topology and write βN \ N = N* (but this does not appear to be standard notation for general 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.
[9] 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.
According to the universal property, there exists a unique extension βa : βN → B.
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).
For any subset, A, of N and a positive integer n in N, we define Given two ultrafilters F and G on N, we define their sum by 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.
