Stone-Čech compacting

The Stone – Čech compactification is a construction of the topology for embedding a topological space in a compact Hausdorff space . The Stone – Čech compactification of a topological space is the “largest” compact Hausdorff space that “contains” as a dense subset . To put it precisely, this means that every mapping from into a compact Hausdorff space can be uniquely factored. If there is a Tychonoff space , then the mapping is an embedding . So you can think of it as a dense subspace of . ${\displaystyle X}$ ${\displaystyle \beta X}$ ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle \beta X}$ ${\displaystyle X}$ ${\displaystyle X\rightarrow \beta X}$ ${\displaystyle X}$ ${\displaystyle \beta X}$

One needs the axiom of choice (e.g. in the form of Tychonoff's theorem ) to show that every topological space has a Stone – Čech compactification. Even for very simple rooms , it is very difficult to get a specific indication of . For example, it is impossible to specify an explicit point from . ${\displaystyle X}$ ${\displaystyle \beta X}$ ${\displaystyle \beta \mathbb {N} \setminus \mathbb {N} }$

The Stone – Čech compactification was found independently by Marshall Stone ( 1937 ) and Eduard Čech ( 1937 ). Čech relied on preliminary work by Andrei Nikolajewitsch Tichonow , who had shown in 1930 that any completely regular room can be embedded in a product of closed intervals . The so-called Stone – Čech compactification is the end of the embedding. Stone, however, looked at the ring of continuous, real-valued functions in a topological space . In its construction, today's Stone – Čech compactification is the set of ultrafilters of a union with a certain topology. ${\displaystyle C(X)}$ ${\displaystyle X}$

Universal property and functoriality

${\displaystyle \beta X}$is a compact Hausdorff space together with a continuous mapping with the following universal property : For every compact Hausdorff space and every continuous mapping there is a uniquely determined, continuous mapping , so that . ${\displaystyle \iota _{X}\colon X\to \beta X}$ ${\displaystyle K}$ ${\displaystyle f\colon X\rightarrow K}$ ${\displaystyle \beta f\colon \beta X\to K}$ ${\displaystyle f=\beta f\circ \iota _{X}}$

The image can be understood intuitively as "embedding" in . is injective if and only if is a complete Hausdorff space , and a topological embedding if and only if is completely regular . In this way of speaking, the figure can be understood as a continuation of to whole . ${\displaystyle \iota _{X}}$ ${\displaystyle X}$ ${\displaystyle \beta X}$ ${\displaystyle \iota _{X}}$ ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle \beta f}$ ${\displaystyle f}$ ${\displaystyle \beta X}$

Since it is a compact Hausdorff space itself, it follows from the universal property that and except for a natural homeomorphism are uniquely determined. ${\displaystyle \beta X}$ ${\displaystyle \beta X}$ ${\displaystyle \iota _{X}}$

• ${\displaystyle \iota _{X}}$is injective if and only if is a complete Hausdorff space . ${\displaystyle X}$
• ${\displaystyle \iota _{X}}$is a topological embedding if and only if is a Tychonoff space. ${\displaystyle X}$
• ${\displaystyle \iota _{X}}$is an open embedding if and only if there is a locally compact Hausdorff space. ${\displaystyle X}$
• ${\displaystyle \iota _{X}}$is a homeomorphism if and only if is a compact Hausdorff space. ${\displaystyle X}$

Some authors assume that the starting room should be a Tychonoff room (or a locally compact Hausdorff room). The Stone – Čech compactification can be constructed for more general spaces, but the figure is no longer embedded if there is no Tychonoff space, because the Tychonoff spaces are precisely the subspaces of the compact Hausdorff spaces. ${\displaystyle \iota _{X}}$ ${\displaystyle X}$

The extension property turns a functor from Top (the category of topological spaces) or Tych (the category of Tychonoff spaces) into CHaus (the category of compact Hausdorff spaces). If we set the inclusion functor from CHaus to Top or Tych , we get that the continuous mappings of ( from CHaus ) are in natural bijection to the continuous mappings (if one considers the restriction to and uses the universal property of ). That is , which means that linksadjungiert to be. ${\displaystyle \beta }$ ${\displaystyle U}$ ${\displaystyle \beta X\to K}$ ${\displaystyle K}$ ${\displaystyle X\to UK}$ ${\displaystyle \iota _{X}(X)}$ ${\displaystyle \beta X}$ ${\displaystyle \operatorname {Hom} (\beta X,K)=\operatorname {Hom} (X,UK)}$ ${\displaystyle \beta }$ ${\displaystyle U}$

Constructions

Construction by means of products

One possibility to produce the Stone-Čech compactification of is to close the image from in ${\displaystyle X}$ ${\displaystyle X}$

${\displaystyle \prod C}$

to take. Here the product is over all images of in compact Hausdorff rooms . However, this is formally not feasible, since the summary of all such images is a real class and not a set, so this product does not even exist. There are several ways you can tweak this idea to make it work. One possibility is to only include those in the product that are defined on a subset of . The cardinality of is greater than or equal to the cardinality of every compact Hausdorff space in which one can map with a dense image. ${\displaystyle X}$ ${\displaystyle C}$ ${\displaystyle C}$ ${\displaystyle {\mathcal {P}}({\mathcal {P}}(X))}$ ${\displaystyle {\mathcal {P}}({\mathcal {P}}(X))}$ ${\displaystyle X}$

Construction with the unit interval

One way to construct is to use the figure ${\displaystyle \beta X}$

${\displaystyle \iota :={\begin{cases}X\to [0,1]^{C(X)}\\x\mapsto (f(x))_{f\in C(X)}\end{cases}}}$

to be used, where is the set of all continuous maps . According to Tychonoff's theorem, it now follows that is compact since is compact. The degree in is therefore a compact Hausdorff room. We show this together with the figure ${\displaystyle C(X)}$ ${\displaystyle X\to [0,1]}$ ${\displaystyle [0,1]^{C(X)}}$ ${\displaystyle [0,1]}$ ${\displaystyle \beta X:={\overline {\iota (X)}}}$ ${\displaystyle [0,1]^{C(X)}}$

${\displaystyle \iota _{X}\colon {\begin{cases}X\to \beta X\\x\mapsto \iota (x)\end{cases}}}$

fulfills the universal property of Stone-Čech compacting.

We consider first . In this case the desired continuation of the projection onto the coordinate in . ${\displaystyle K=[0,1]}$ ${\displaystyle f\colon X\rightarrow [0,1]}$ ${\displaystyle f}$ ${\displaystyle [0,1]^{C(X)}}$

If any compact Hausdorff space is, according to the above construction, it is homeomorphic to . The injectivity of the embedding follows from the lemma of Urysohn , the surjectivity and the continuity of the inverses from the compactness of . It is now sufficient to continue component by component. ${\displaystyle K}$ ${\displaystyle \beta K}$ ${\displaystyle K}$ ${\displaystyle f\colon X\rightarrow Y\cong \beta Y\subset [0,1]^{C(Y)}}$

The universal property of the unit interval required in this proof 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 such that and are different. Instead , one could have used any cogenerator or any cogenerating amount. ${\displaystyle f,g\colon A\rightarrow B}$ ${\displaystyle h\colon B\rightarrow [0,1]}$ ${\displaystyle h\circ f}$ ${\displaystyle h\circ g}$ ${\displaystyle [0,1]}$

Construction using ultrafilters

Discreet spaces

Is discreet, then you can as the set of all ultrafilters on the topology Stone construct. The embedding of is then done by identifying the elements from with the one-point filters. This construction complies with Wallman compacting for discrete rooms . ${\displaystyle X}$ ${\displaystyle \beta X}$ ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle X}$

Again one has to check the universal property: be an ultrafilter . Then for each image , with compact Hausdorff space, an ultrafilter on . This ultrafilter has a clear limit because it is compact and Hausdorff. Now one defines and one can show that this is a continuous continuation of . ${\displaystyle F}$ ${\displaystyle X}$ ${\displaystyle f\colon X\rightarrow K}$ ${\displaystyle K}$ ${\displaystyle f(F)}$ ${\displaystyle K}$ ${\displaystyle x\in K}$ ${\displaystyle K}$ ${\displaystyle \beta f(F)=x}$ ${\displaystyle f}$

Equivalently, one can take the Stone space of the complete Boolean algebra of all subsets of as the Stone-Čech compactification. This is really the same construction, since the stone space of this Boolean algebra is the set of ultrafilters, or equivalent of the prime ideals (or homomorphisms in the two-element Boolean algebra) of Boolean algebra, which is the same as the set of ultrafilters . ${\displaystyle X}$ ${\displaystyle X}$

General Tychonoff Rooms

If there is any Tychonoff space, then instead of all subsets only the -sets of are used to obtain the connection with the topology: ${\displaystyle X}$ ${\displaystyle z}$ ${\displaystyle X}$

If a continuous function, then the z-set is called by . The set of all -sets of is denoted by, i. H. . ${\displaystyle f\colon X\to \mathbb {R} }$ ${\displaystyle \{x\in X:f(x)=0\}}$ ${\displaystyle f}$ ${\displaystyle Z(X)}$ ${\displaystyle z}$ ${\displaystyle X}$ ${\displaystyle Y\in Z(X)\Leftrightarrow \exists f\in C(X,\mathbb {R} )\left(f(x)=0\Leftrightarrow x\in Y\right)}$

The sets are ordered by the subset relation and you can define filters as usual. An -Ultrafilter is a maximal filter. ${\displaystyle z}$ ${\displaystyle z}$

If we denote by the set of all Ultra-filter with the topology defined by the quantities of generated, then: Since for each point of because of Tychonoff property one is quantity, is a -Ultrafilter. Therefore, the picture with an embedding. One then shows that the constructed space is a compact Hausdorff space and that the image is close up in it. This is true, ultimately, from the fact that any limited function can be continued. ${\displaystyle J(X)}$ ${\displaystyle Z(X)}$ ${\displaystyle z}$ ${\displaystyle X}$ ${\displaystyle x\in X}$ ${\displaystyle \{x\}}$ ${\displaystyle z}$ ${\displaystyle Z(x):=\{Y\in Z(X)\mid x\in Y\}}$ ${\displaystyle z}$ ${\displaystyle \iota \colon X\to J(X)}$ ${\displaystyle \iota (x)=Z(x)}$ ${\displaystyle X}$ ${\displaystyle J(X)=\beta X}$

Construction using C * algebras

If a completely regular room is, one can identify the Stone-Čech compactification with the spectrum of . Here stands for the unital commutative C * -algebra of all continuous and bounded mappings with the supremum norm . The spectrum is the set of multiplicative functionals with the subspace topology of the weak - * - topology of the dual space of , note . For each there is a multiplicative functional. If one identifies with , one obtains , and one can show that it is homeomorphic to Stone – Čech compactification . ${\displaystyle X}$ ${\displaystyle C_{b}(X)}$ ${\displaystyle C_{b}(X)}$ ${\displaystyle X\rightarrow \mathbb {C} }$ ${\displaystyle {\tilde {X}}}$ ${\displaystyle C_{b}(X)}$ ${\displaystyle {\tilde {X}}\subset C_{b}(X)'}$ ${\displaystyle x\in X}$ ${\displaystyle \delta _{x}\colon C_{b}(X)\rightarrow \mathbb {C} ,\,\delta _{x}(f):=f(x)}$ ${\displaystyle x\in X}$ ${\displaystyle \delta _{x}\in {\tilde {X}}}$ ${\displaystyle X\subset {\tilde {X}}}$ ${\displaystyle {\tilde {X}}}$ ${\displaystyle \beta X}$

See also

• Compactification
• Notation set for Boolean algebras

literature

• Boto von Querenburg : Set theoretical topology (= Springer textbook ). 3rd, revised and expanded edition. Springer, Berlin et al. 2001, ISBN 3-540-67790-9 .
• Karsten Evers: Set theoretical topology. 2011. (PDF, 1.72 MB).

Individual evidence

1. ^ Boto von Querenburg: Set theoretical topology. 2001, p. 334.
2. ^ Nicolas Bourbaki : Eléments de mathématique. Topology Générale. Ch. 1/4 . Springer, Berlin et al. 2007, ISBN 978-3-540-33936-6 , chap. 9 , p. 10 .
3. ^ See Leonard Gillman, Meyer Jerison: Rings of Continuous Functions. van Nostrand Reinhold, New York NY et al. 1960, § 8, for the entire construction.
4. ^ John B. Conway: A Course in Functional Analysis. Springer-Verlag, 2007, ISBN 978-0-387-97245-9 , pp. 137f. ( limited preview in Google Book search).