Stone–Čech compactification

The Stone - Čech compactification is a construction of the topology for the embedding of a topological space X into a compact Hausdorff space. The Stone - Čech compactification of a topological space X is the " largest " compact Hausdorff space, the "contains" as a dense subset. Precise terms this means that any map from respect uniquely factored into a compact Hausdorff space. If X is a Tychonoff space, then the mapping is an embedding. So you can imagine as a dense subspace of.

You need the axiom of choice (in the form of the set of Tychonoff ) to show that every topological space has a Stone - Čech compactification. Even for very simple spaces, it is very hard to get a concrete indication of. For example, it is impossible to specify an explicit point.

The Stone - Čech compactification was by Marshall Stone ( 1937) and Eduard Čech ( 1937) found independently. Čech was based on previous work of Andrei Nikolaevich Tikhonov, who showed in 1930 that every completely regular space can be embedded into a product of closed intervals. The Stone - Čech compactification today is then the completion of the embedding. Stone, however, looked at the ring of continuous, real-valued functions on a topological space. Its design today Stone - Čech compactification is the set of ultrafilters of an association with a particular topology.

Universal property and functoriality

Is a compact Hausdorff space together with a continuous map with the following universal property: for every compact Hausdorff space and every continuous map, there exists a unique, continuous map, so.

The figure can be interpreted intuitively as " embedding " of in. is injective when a complete Hausdorff space, and just then a topological embedding if it is completely regular. The figure can be interpreted in this way of speaking as a continuation of completely.

Since is a compact Hausdorff space itself, it follows from the universal property that and are uniquely determined up to a natural homeomorphism.

• Is injective when a complete Hausdorff space.
• Is a topological embedding if and only if a Tychonoff space.
• Is exactly then an open embedding, if a locally compact Hausdorff space.
• Is a homeomorphism if and only if a compact Hausdorff space.

Some authors assume that the output space is to be a Tychonoff space ( or even a locally compact Hausdorff space ). The Stone - Čech compactification can be constructed for general spaces, but the picture is no more, if no space is Tychonoff embedding, because the Tychonoff spaces are just the subspaces of compact Hausdorff spaces.

The extension property makes a functor from Top ( the category of topological spaces ) or Tych ( the category of Tychonoff spaces ) in Chaus ( the category of compact Hausdorff spaces). If we set the Inklusionsfunktor of Chaus top or Tych, we get that the continuous mappings of (K from Chaus ) are in natural bijection with the continuous maps ( If one considers the restriction on and the universal property of used). That is, which means that linksadjungiert is to U.

Constructions

Construction with products

One way the Stone - Čech compactification to be generated by X, in completing the image of X in

To take. Here the product is over all images of X into compact Hausdorff spaces is. However, this is formally not feasible, as the summary of all such mappings is a proper class and not a lot, so this product does not exist. There are several ways this idea to change so that it works. For example, you can include only those in the product that are defined on a subset of the. The cardinality of is greater than the cardinality is equal to every compact Hausdorff space in which one can represent X with dense image.

Construction with the unit interval

One way to construct, is the picture

To use, wherein the set of pictures is constant. By the theorem of Tychonoff now follows that is compact, since is compact. The conclusion is therefore in a compact Hausdorff space. We show that this, together with the picture

Satisfies the universal property of the Stone - Cech compactification.

We consider first. In this case, the continuation of the desired projection to the coordinate is in the.

Is an arbitrary compact Hausdorff space, it is by the above construction homeomorphic to. The injectivity of the embedding follows from the lemma of Urysohn here, surjectivity and continuity of the inverse of the compactness of. Now, you just componentwise continue.

The required in this proof universal property of the unit interval is that it is a cogenerator of the category of compact Hausdorff spaces. This means that there is a morphism for any two different morphisms, and so are different. Instead one would therefore can use any cogenerator or any kogeneriende amount.

Structure by means of ultrafilters

Discrete spaces

Is discrete, then you can as the set of all ultrafilters on construct with the Stone topology. The embedding is then carried out by identifying the elements of the Einpunktfiltern. This construction is true for discrete spaces with the Wallman compactification match.

Again, you have to check the universal property: Let an ultrafilter on. Then, for each image, with a compact Hausdorff space, an ultrafilter on. This ultrafilter has a unique limit, because it is compact and Hausdorff. Now you defined and one can show that this is a continuous extension of.

Equivalently, one can take the Stone space of the complete Boolean algebra of all subsets of Stone - Čech as the compactification. This is really the same construction as the Stone space of the Boolean algebra the set of ultrafilters or equivalent to the prime ideals is the Boolean algebra (or homomorphisms in the two-element Boolean algebra), which is the same as the set of ultrafilters on.

General Tychonoff spaces

Is any Tychonoff space, is taken instead of all subsets, only the amounts of z in order to obtain the connection with the topology:

The z- levels are ordered by the subset relation, and can be defined as common filter. A z- ultrafilter is a maximal filter.

Refers to the set of all ultra- filter to the topology, which is produced by the quantities of Z, is given by: Since, for each point of due to the Tychonoff property is a z- amount is a z- ultrafilter. Therefore, the figure with an embedding. One then still shows that the constructed space is a compact Hausdorff space and that the image of is dense in it. That is true, finally follows from the fact that every bounded function can be continued.

Design by C *-algebras

If a completely regular space, one can identify the Stone - Čech compactification with the spectrum of. Here is *-algebra of all continuous and bounded mappings with the supremum norm for the commutative C. The range is the set of multiplicative functional with the partial space of the low-level topology * topology of the dual space of note. For each is a multiplicative functional. If we identify with, we get, and one can show that compactification is homeomorphic to the Stone - Čech.