Compactification (mathematics)

Compactification is a generic term from the mathematical branch of topology. Under a compactification is understood to be the assignment of compact spaces at certain topological spaces, so that the respective associated compact space, the compactification of the original space, topological properties of the original space takes over. In many cases, the original space will be considered as part of the room space compactified.

• 4.1 Gelfand - Kolmogorov

Usual receivables

• The space is homeomorphic to a subspace of the compactification, which is equivalent to the fact that an embedding exists in the compactification, ie an injective, continuous and relatively open figure.
• Nestled construed in the compactification, it is a dense subset of these, to guarantee the uniqueness of sequels of continuous maps on the compactification (see below).
• As large as possible classes of continuous maps on the space can be steadily continue on the compactification or at least similarly transferred to the compactification.
• The compactification satisfies the Hausdorf prickly stem.

Examples

Generally, there is a room for many different compactifications, which in part is differ dramatically.

Stone - Čech compactification

Every completely regular space can be compactified by the Stone - Čech compactification. There are a number of different constructions and the resulting space has many attributes that make him such

• Is, if this federation belonging, a maximum of the Association of compactifications of which contain as dense subspace
• Every bounded function can be continued after

Einpunktkompaktifizierung ( Alexandroff compactification )

The Russian mathematician Paul Alexandroff has given a construction that leads to an arbitrary topological space into a compact extension:

It will add a single new point taken. The topology, that is the open portion of quantities is then of the given amount of the open part and the complements of the completed compact amounts which are in the.

The embedding is called Alexandroff extension or Alexandroff compactification of. She has most of the properties required above. But where: is exactly then a Hausdorff space if it is locally compact and Hausdorff. In particular, for locally compact Hausdorff spaces normally ( like any compact Hausdorff space ) and thus by the lemma of Urysohn completely regular, which carries over to the original space: Every locally compact Hausdorff space is completely regular.

Concrete examples

• The Einpunktkompaktifizierung of real numbers is topologically the same structure of a circle, ie a. The Einpunktkompaktifizierung of complex numbers is the Riemann sphere, the structure of the surface of a sphere, so a corresponding. In general, the Einpunktkompaktifizierung of homeomorphic to the n-dimensional sphere
• While the Einpunktkompaktifizierung the set of natural numbers really only one more point ( " infinity " ) that Stone - Čech compactification has the power.
• Is also Alexandroff compactification and Stone - Čech compactification for the first uncountable ordinal with the order topology.

Continuability continuous functions

Important for the applicability of compactifications is also the possibility of continuous functions on the continuability to kompaktifizierenden space on the compactification. About the behavior of continuous functions on compact spaces be easier to describe and then transferred to the restriction of the function to the original space. In addition, preserved under compactification also universal properties of the room. The demand for tightness of the original space in the compactification guaranteed if the compactification is Hausdorff, the uniqueness of the continuation.

On a locally compact Hausdorff space can be exactly the continuous functions continue steadily to a function on the Einpunktkompaktifizierung that clearly spoken " at infinity seek a fixed value ", with continuous real functions, for example, those which " vanish at infinity ", ie their value certain distances from the origin arbitrarily close to the unit zero, these are the C0 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 Tikhonov - space all continuous functions in a compact Hausdorff space allowed to continue steadily on the compactified space, such as in the case of real-valued functions, all bounded continuous functions.

The continuity of functions into a space is preserved when they perceives as functions in the compactified space, if a continuous and injective embedding in the compactified space exists.

Application

Many sets of the topology are first proved for compact spaces, since the finiteness condition are easier to lead evidence ( in its various formulations). As a further step is then tried for other rooms to construct a suitable compactification and to see under what conditions they can transfer results. As an example of application, we consider the set of Gelfand - Kolmogorov:

Gelfand - Kolmogorov

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

Be the ring of continuous functions from to ( defined with pointwise addition and multiplication ) and the subring of bounded functions.

• ( Gelfand - Kolmogorov ): In every Tychonoff space, there is a 1-1 correspondence between the maximal ideals of the and. In both cases "fixed" every maximal ideal of exactly one point.

More precisely: in there for each maximal ideal ( exactly ) one point, the continuous extension of after is.

For reads the description for maximal ideals, is where and to the financial statements.

Related terms

Analogous to the notion of compactification one can proceed with compact related terms with most: the term Pseudokompaktifizierung are obtained for example by replacing compact in the definition of pseudo- compact.