Compact space

Compactness is a central concept of mathematical topology, namely a property which belongs to a topological space or not. It is assumed in many mathematical statements - often in a weaker form than Lindelöf property or para- compactness. Local compactness is also a weakened condition in the case of Hausdorff spaces. A compact set is called depending on the context or even compact set compact space; there is not much difference if it is a subset of an upper chamber or not.

Simple examples of compact sets are closed and bounded subsets of Euclidean space as the interval ( in ). Simple counter-examples are the non-compact quantities or.

• 4.1 Compact spaces
• 4.2 Non- compact spaces

Definition

Compactness in Euclidean space

A subset of the Euclidean space is called compact if it is closed and bounded. For this particular definition of the set of Heine- Borel applies:

The set of Heine- Borel motivates the following generalization of the definition of compactness in topological spaces.

Compactness in topological spaces

A topological space is called compact if every open cover

A finite subcover

Possesses.

A subset of a topological space is called compact if every open cover

A finite subcover

Possesses. The two concepts are compatible. A subset of a topological space is compact if it is compact as a topological space with the subspace topology.

Some authors such as Nicolas Bourbaki use for the property as defined herein the term quasi- compact and to reserve the term "compact" for compact Hausdorff spaces; This is common by the French influence in particular in algebraic geometry.

History

Around the year 1900, the following characterizations of compact subsets were the well-known:

The first characterization is dependent on the selected metric. The other three, however, characterization can be applied to arbitrary topological spaces and thus offer a possibility for a compact concept to define topological spaces. Maurice René Fréchet called 1906 subsets of metric spaces compact, met the second property. This definition was later transferred to topological spaces. So you called in the modern sense countably compact spaces then compact. Pavel Sergeyevich Alexandrov and Pavel Urysohn Samuilowitsch led in 1924 to today's compactness term within the meaning of the fourth property a. Spaces that satisfy this property, they called bicompact. This compactness definition sat down, however, until about 1930, when Andrei Nikolaevich Tikhonov proved that any products bicompact spaces arise bicompact rooms again. This result is now known as the set of Tychonoff. This does not apply countably compact and sequentially compact spaces ( property three).

From finite to compact

An important reason for the observation of compact spaces is that they can be in some ways seen as a generalization of finite topological spaces, in particular all finite spaces are compact. There are many results, which can be easily proved for finite sets whose evidence must then be transferred with small changes on compact spaces. Here is an example:

We assume that a Hausdorff space is a point and a finite subset of which does not contain. Then we can, separated by environments: are for each and disjoint neighborhoods, each containing respectively. Then the intersection of all and the union of all the required environments and.

Is not finite, the proof is no longer valid, since the average of infinitely many environments must be no longer around. For the case that is compact, the proof idea can but transferred as follows:

We assume again that a Hausdorff space is a point and a compact subset of, which does not contain. Then we can, separated by environments: are for each and disjoint open neighborhoods, each containing respectively. As is compact and is covered by the open volume, there is finally with many points. Then the intersection of all and the union of all, the required environments and.

One can see from this example how the compactness is used to get from possibly infinitely many environments on a finite number, with which then the well-known proof of finite sets can be continued. Many proofs and theorems on compact sets follow this pattern.

Examples

Compact spaces

• Considering the closed unit interval [0, 1] as a subset of the default topology is provided with, the interval is a compact, topological space. Also compact the balls and spheres are considered as subsets of the provided with the standard topology for arbitrary natural numbers.
• All topological spaces with finite topology, such as finite spaces are compact.
• For a natural number, consider the set of all sequences of values. On this amount, one can define a metric by setting, where. If, as is. From the set of Tychonoff (see below ) it follows that the induced by this metric topological space is compact. This construction can be performed for each of a finite set, not only for. The resulting metric space is even ultrametrisch. The following applies: Is, then the mapping is a homeomorphism of the Cantor set.
• Is a prime number, then the mapping is a homeomorphism of the -adic integers.

Non- compact spaces

• The real numbers equipped with the standard topology is not compact. Also not compact are the half-open interval, the integers or the natural numbers considered as subsets of. However, you know, for example, with the trivial topology, so is compact. Whether a set is compact, therefore depends in general on the selected topology.
• The closed unit ball of the space of bounded real number sequences (see Lp space) is not compact, even though it is closed and bounded. It is generally accepted that the unit ball is compact in a Banach space if the dimension of the space is finite.

Properties

• The image of a compact set under a continuous function is compact. Consequently, a real-valued continuous function on a non-empty compact set to a global minimum and a global maximum.

• A continuous function on a compact metric space is uniformly continuous. This statement is also known as a set of Heine.

• Each environment of a compact set in a uniform space is uniform environment. That is, it is a neighborhood in the area. In the metric case, this means that all points are equally-sized balls of a size selected within the environment. The neighborhood can even be chosen so that the complement of the area is the vicinity of outside of the compact set with the neighborhood.

• Every infinite sequence of elements of a compact set has an accumulation point in. Complete the first countability axiom, then there exists even in a convergent subsequence. However, the converse is not true in any topological space, ie a subset in which each episode one ( in the subset ) convergent subsequence has (such a subset is called sequentially compact, see below), does not have to be compact. ( An example is the set of countable ordinals with the order topology. )

• A closed subset of a compact space is compact.

• A compact subset of a Hausdorff space is closed.

• A non-empty compact subset of the real numbers has a largest and a smallest element (see also supremum, infimum ).

• For each subset of Euclidean space, the following three statements are equivalent ( compare theorem of Heine - Borel ):   is compact, ie every open cover has a finite subcover.
• Each sequence in the set has a convergent subsequence in ( ie at least one accumulation point).
• The set is closed and bounded.

• A metric space is compact if it is limited completely and totally.

• The product of any class of compact spaces is compact in the product topology. ( Set of Tychonoff - this is equivalent to the axiom of choice )

• A compact Hausdorff space is normal.

• Every continuous bijection from a compact space to a Hausdorff space is a homeomorphism.

• A metric space is compact if every sequence in the space has a convergent subsequence with limit her in the room.

• A topological space is compact if every net in the room has a subnet that has a limit in the space.

• A topological space is compact if every filter has on the space a convergent refinement.

• A topological space is compact if every ultrafilter converges to the room.

• A topological space can be accurately then embedded in a compact Hausdorff space, if it is a Tychonoff space.

• Every topological space is a dense subspace of a compact space which has at most one point more than. ( See also Alexandroff compactification. )

• A metrizable space is compact if every metric space homeomorphic to complete.

• If the metric space is compact and an open cover of is given, then there exists a number such that every subset of included with diameter in a element of the coverage is. ( Lemma of Lebesgue )

• Every compact Hausdorff space can be precisely to a uniform structure, which induces the topology. The converse is not true.

• If a topological space has a sub-base, so that every cover of the space by elements of the sub-base has a finite subcover, then the space is compact. ( Alexander's sub-base - set)

• Two compact Hausdorff spaces and are homeomorphic if and only if their rings of continuous real-valued functions and are isomorphic.

Other forms of compactness

There are some topological properties which are equivalent to compact metric space, but not in general equivalent topological areas:

• Follow- compact: Every sequence has a convergent subsequence.

• Countably compact: Every countable open cover has a finite subcover. ( Or, equivalently, every infinite subset has a limit point -. )

• Pseudo Compact: Each real-valued continuous function on the space is limited.

• Weak countably compact: Every infinite subset has a limit point.

While these concepts for metric spaces are equivalent, there are generally the following relations:

• Compact spaces are countably compact.
• Consequence Compact spaces are countably compact.
• Countably compact spaces are pseudo- compact and weakly countably compact.