Compactification

This article deals with compactification in terms of mathematics. For compactification in terms of string theory, see Kaluza-Klein compactification .

Compactification is a generic term from the mathematical sub-area of topology . Compactification is understood to mean the assignment of compact spaces to certain topological spaces, so that the respectively assigned compact space, the compactification of the original space, takes on topological properties of the original space. In many cases the original space can be understood as a subspace of the compacted space.

Usual demands

• The space is homeomorphic to a subspace of the compactification, which is equivalent to an embedding in the compactification, that is, an injective , continuous and relatively open mapping.
• Understood embedded in the compactification, it is a dense subset of this, this guarantees the uniqueness of continuations of continuous mappings on the compactification (see below).
• The largest possible classes of continuous mappings in space can be continuously continued on the compactification or at least transferred in a similar way to the compactification.
• The compacting fulfills the Hausdorff property .

Examples

In general, there are many different compactifications for a room, e.g. T. differ dramatically.

Stone-Čech compacting

→ Main article : Stone-Čech compacting

Any completely regular room can be compacted by Stone-Čech compacting . There are a number of different constructions for this and the resulting space has many characteristics that distinguish it, e.g. B. ${\displaystyle \beta X}$

• ${\displaystyle \beta X}$If it belongs to this association, it is at most in the association of compactifications, which contain as a dense subspace ${\displaystyle X}$
• any restricted function can be continued after ${\displaystyle f\colon X\to \mathbb {R} }$ ${\displaystyle \beta X}$

One-point compactification (Alexandroff compactification)

→ Main article : Alexandroff compactification

The Russian mathematician Paul Alexandroff has given a construction which leads to a compact extension for any topological space : ${\displaystyle X}$

A single new point is added. The topology, i.e. the open subsets of , then consists of the given open subsets of and the complements of the closed, compact sets that lie in. ${\displaystyle \omega }$ ${\displaystyle X}$ ${\displaystyle X^{*}=X\cup \{\omega \}}$ ${\displaystyle X}$ ${\displaystyle X}$

The embedding is called Alexandroff expansion or Alexandroff compactification of . It has most of the properties required above. The following applies: is a Hausdorff space if and only if it is locally compact and Hausdorff. In particular, it is normal for locally compact Hausdorff spaces (like every compact Hausdorff space) and thus completely regular according to Urysohn's lemma , which is carried over to the original space : Every locally compact Hausdorff space is completely regular. ${\displaystyle \varphi \colon X\rightarrow X^{*}}$ ${\displaystyle X}$ ${\displaystyle X^{*}}$ ${\displaystyle X}$ ${\displaystyle X^{*}}$ ${\displaystyle X}$

Concrete examples

• The one-point compactification of the real numbers corresponds topologically to the structure of a circle, i.e. a . The one-point compactification of the complex numbers is the Riemann number sphere , the structure of which corresponds to the surface of a sphere, i.e. a 2-sphere . In general, the one-point compactification of the homeomorphic to the n-dimensional sphere is . ${\displaystyle \mathbb {R} }$ ${\displaystyle S^{1}}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle S^{2}}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle S^{n}}$
• While the one-point compactification of the set of natural numbers actually only contains one further point (countable "infinite"), the Stone-Čech compactification has the power with the help of the validity of the continuum hypothesis . ${\displaystyle \mathbb {N} }$ ${\displaystyle \aleph _{0}=Card(\omega )}$ ${\displaystyle \beth _{2}=2^{2^{\aleph _{0}}}=2^{\aleph _{1}}=\aleph _{2}}$
• For the first uncountable ordinal number with the order topology , Alexandroff compactification and Stone-Čech compactification are at the same time. ${\displaystyle \omega _{1}}$ ${\displaystyle \omega _{1}+1}$

Continuability of continuous functions

The possibility of continuation of continuous functions in the space to be compacted to the compactification is also important for the applicability of compactifications. For example, the behavior of continuous functions in compact spaces can be easier to describe and then transferred to the restriction of the function in the original space. In addition, universal properties of the room can also be retained with compacting. The requirement for tightness of the original space in the compactification guarantees, if the compactification is Hausdorffsch , the uniqueness of the continuation.

On a locally compact Hausdorff space, precisely the continuous functions can be continuously continued to form a function on the one-point compactification, which, clearly speaking, “strive for a fixed value at infinity”, with continuous real functions for example those that “disappear at infinity”, i.e. their value If it comes close to zero at a certain distance from the origin, these are the C ₀ functions . Generally speaking, the image of the filter base of the complements of compact sets converges. In the case of the Stone-Čech compactification of a Tichonow space, all continuous functions in a compact Hausdorff space can be continuously continued in the compacted space, so for example all limited continuous functions in the case of real-valued functions.

The continuity of functions in a space is preserved if one understands them as functions in the compactified space, if a continuous and injective embedding in the compactified space exists.

application

Many theorems of topology are first proved for compact spaces, since the finiteness condition (in its various formulations) makes it easier to give proofs. As a further step, an attempt is then made to construct a suitable compactification for other rooms and to see under which conditions results can be transferred. As an example of an application, consider the Gelfand-Kolmogoroff theorem:

Gelfand-Kolmogoroff's theorem

This sentence is an example of the fact that statements about a room can be obtained directly with the help of Stone-Čech compactification.

${\displaystyle \textstyle {\mathcal {C}}(X)}$be the ring of continuous functions from to (with pointwise defined addition and multiplication) and the subring of bounded functions. ${\displaystyle X}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \textstyle {\mathcal {C}}^{*}(X)}$

• ( Gelfand - Kolmogoroff ) : In every Tychonoff space there is a 1-1 assignment between the maximum ideals of and of . In both cases each maximal ideal "fixes" exactly one point . ${\displaystyle X}$ ${\displaystyle \textstyle {\mathcal {C}}^{*}(X)}$ ${\displaystyle \textstyle {\mathcal {C}}(X)}$ ${\displaystyle p\in \beta (X)}$

More precisely: in there is (exactly) a point with for every maximal ideal , where is the continuous continuation of to . ${\displaystyle \textstyle {\mathcal {C}}^{*}(X)}$ ${\displaystyle \textstyle I_{p}}$ ${\displaystyle p\in \beta (X)}$ ${\displaystyle \textstyle I_{p}=\left\{f\in {\mathcal {C}}^{*}(X)|f^{\beta }(p)=0\right\}}$ ${\displaystyle f^{\beta }}$ ${\displaystyle f}$ ${\displaystyle \beta X}$

For , the corresponding description for maximum ideals is:, where and stands for the degree in . ${\displaystyle \textstyle {\mathcal {C}}(X)}$ ${\displaystyle \textstyle I_{p}=\left\{f\in {\mathcal {C}}(X)|p\in cl_{\beta X}Z_{X}(f)\right\}}$ ${\displaystyle Z_{X}(f)=\left\{x\in X|f(x)=0\right\}}$ ${\displaystyle cl_{\beta X}}$ ${\displaystyle \beta X}$

Related terms

Analogous to the idea of ​​compactification, one can proceed with most of the terms that are related to compactly : The term pseudocompactification is obtained, for example, by replacing compact with pseudocompact in the definition .

Individual evidence

1. ↑ on the whole for this example: L. Gillman, M. Jerison: Rings of Continuous Functions. 1976, chap. 6 f.

literature

• Paul Alexandroff : About the Metrisation the compact retail topological spaces. In: Mathematical Annals . Vol. 92, No. 3/4, 1924, pp. 294-301, doi : 10.1007 / BF01448011 , digitized version (PDF; 646 kB) .
• Leonard Gillman, Meyer Jerison: Rings of Continuous Functions (= Graduate Texts in Mathematics. Vol. 43). Springer, Berlin et al. 1976, ISBN 3-540-90198-1 .
• Boto von Querenburg : Set theoretical topology (= Springer textbook ). 3rd, revised and expanded edition. Springer, Berlin et al. 2001, ISBN 3-540-67790-9 .