Compact space

Compactness is a central concept in mathematical topology , a property that a topological space has or does not have. It is assumed in many mathematical statements - often in a weakened form as Lindelöf property or paracompactness . Local compactness is also a weakened requirement in the case of Hausdorff rooms . A compact set is also called a compact or compact space , depending on the context ; it is irrelevant whether it is a subset of an upper space.

Simple examples of compact sets are closed and bounded subsets of Euclidean space like the interval . Simple counterexamples are the non-compact sets (not restricted) or (not closed). ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle [0,1]\subset \mathbb {R} }$ ${\displaystyle \mathbb {N} \subset \mathbb {R} }$ ${\displaystyle \left[0,1\right[\subset \mathbb {R} }$

definition

Compactness in Euclidean space

→ Main article : Compactness (real numbers)

A subset of Euclidean space is called compact if it is closed and bounded . For this particular definition, Heine-Borel's theorem applies : ${\displaystyle \mathbb {R} ^{n}}$

A subset of is compact if and only if every open coverage of the subset contains a finite partial coverage . ${\displaystyle \mathbb {R} ^{n}}$

Heine-Borel's theorem motivates the following generalization of the definition of compactness to topological spaces.

Compactness in topological spaces

A topological space is called compact if every open cover ${\displaystyle (X,{\mathcal {T}})}$

${\displaystyle X=\bigcup _{i\in I}U_{i}\quad {\textrm {mit}}\quad U_{i}\in {\mathcal {T}}}$

a finite partial coverage

${\displaystyle X=U_{i_{1}}\cup U_{i_{2}}\cup \dotsb \cup U_{i_{n}}{\text{ mit }}i_{1},\dotsc ,i_{n}\in I}$

owns.

A subset of a topological space is called compact if every open cover ${\displaystyle M}$ ${\displaystyle (X,{\mathcal {T}})}$

${\displaystyle M\subseteq \bigcup _{i\in I}U_{i}\quad {\textrm {mit}}\quad U_{i}\in {\mathcal {T}}}$

a finite partial coverage

${\displaystyle M\subseteq U_{i_{1}}\cup U_{i_{2}}\cup \dotsb \cup U_{i_{n}}{\text{ mit }}i_{1},\dotsc ,i_{n}\in I}$

owns. The two terms are compatible. A subset of a topological space is compact if and only if it is compact as a topological space with the subspace topology .

Some authors, such as Nicolas Bourbaki , use the term quasi-compact for the property defined here and reserve the term compact for compact Hausdorff spaces . Some authors call the compactness for a clearer definition of the sequential compactness also cover compactness .

history

Around 1900 the following characterizations of compact subsets of the were known: ${\displaystyle A}$ ${\displaystyle \mathbb {R} ^{n}}$

1. The subset is bounded and closed. ${\displaystyle A}$
2. Every subset of with infinitely many elements has at least one accumulation point . ( Theorem by Bolzano-Weierstrass ) ${\displaystyle A}$
3. Each sequence in has a sub-sequence that converges in. ( Theorem by Bolzano-Weierstrass ) ${\displaystyle A}$ ${\displaystyle A}$
4. Every open cover of has a finite partial cover . ( Heine-Borel's theorem ) ${\displaystyle A}$

The first characterization depends on the chosen metric . The other three characterizations, on the other hand, can be transferred to any topological spaces and thus offer the possibility of defining a concept of compactness for topological spaces. In 1906, Maurice René Fréchet called subsets of metric spaces compact that fulfilled the second property. This definition was later transferred to topological spaces. So the countable compact spaces in today's sense were called compact back then. Pawel Sergejewitsch Alexandrow and Pawel Samuilowitsch Urysohn introduced the current concept of compactness in the sense of the fourth property in 1924. She called spaces that fulfilled this characteristic bicompact. However, this definition of compactness did not gain acceptance until 1930, when Andrei Nikolajewitsch Tichonow proved that any product of bicompact rooms could result in bicompact rooms. This result is known today as the Tychonoff Theorem . This does not apply to countably compact and sequentially compact spaces (property three).

From finiteness to compactness

The point is separated from. ${\displaystyle x}$ ${\displaystyle A=\{a,b,c\}}$

An important reason for considering compact spaces is that in some respects they can be seen as a generalization of finite topological spaces; in particular, all finite spaces are also compact. There are many results that can easily be proved for finite sets, the proofs of which can then be transferred to compact spaces with small changes. Here's an example:

We assume that is a Hausdorff space , a point from and a finite subset of that does not contain. Then we can and by environments separate: for each of are and disjoint neighborhoods, each respectively contain. Then the intersection of all and the union of all are the required neighborhoods of and . ${\displaystyle X}$ ${\displaystyle x}$ ${\displaystyle X}$ ${\displaystyle A}$ ${\displaystyle X}$ ${\displaystyle x}$ ${\displaystyle x}$ ${\displaystyle A}$ ${\displaystyle a}$ ${\displaystyle A}$ ${\displaystyle U_{a}(x)}$ ${\displaystyle V(a)}$ ${\displaystyle x}$ ${\displaystyle a}$ ${\displaystyle U_{a}(x)}$ ${\displaystyle V(a)}$ ${\displaystyle x}$ ${\displaystyle A}$

If it is not finite, the proof is no longer valid, since the intersection of infinitely many environments no longer has to be an environment. For the case that is compact, the idea of ​​proof can be transferred as follows: ${\displaystyle A}$ ${\displaystyle A}$

We again assume that a Hausdorff space is a point from and a compact subset of that does not contain. Then we can and by environments separate: for each of are and disjoint open neighborhoods, each respectively contain. Since is compact and is covered by the open sets , there are finitely many points with . Then the intersection of all , and the union of all , who needed environments and . ${\displaystyle X}$ ${\displaystyle x}$ ${\displaystyle X}$ ${\displaystyle A}$ ${\displaystyle X}$ ${\displaystyle x}$ ${\displaystyle x}$ ${\displaystyle A}$ ${\displaystyle a}$ ${\displaystyle A}$ ${\displaystyle U_{a}(x)}$ ${\displaystyle V(a)}$ ${\displaystyle x}$ ${\displaystyle a}$ ${\displaystyle A}$ ${\displaystyle V(a)}$ ${\displaystyle a_{1},\ldots ,a_{n}\in A}$ ${\displaystyle A\subseteq V(a_{1})\cup \ldots \cup V(a_{n})}$ ${\displaystyle U_{a_{i}}(x)}$ ${\displaystyle V(a_{i})}$ ${\displaystyle i=1,\ldots ,n}$ ${\displaystyle x}$ ${\displaystyle A}$

This example shows how compactness is used to get from possibly infinitely many neighborhoods to finitely many, with which the well-known proof for finite sets can then be continued. Many proofs and theorems about compact sets follow this pattern.

Examples

Compact spaces

• If you consider the closed unit interval as a subset of provided with the standard topology , the interval is a compact, topological space. The - spheres and - spheres are also compact, viewed as subsets of the provided with the standard topology for any natural numbers . ${\displaystyle [0,1]}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle n}$ ${\displaystyle n-1}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle n}$
• All topological spaces with finite topology, e.g. B. finite spaces are compact.
• For a natural number consider the set of all sequences with values . A metric can be defined on this set by setting, where . Is so be . From Tychonoff's theorem (see below) it follows that the topological space induced by this metric is compact. This construction can be done for any finite set, not just for . The resulting metric space is even ultrametric . The following applies: ${\displaystyle p>1}$ ${\displaystyle p^{\mathbb {N} }}$ ${\displaystyle \{0,\dotsc ,p-1\}}$ ${\displaystyle d}$ ${\displaystyle d((x_{k}),(y_{k})):=p^{-m}}$ ${\displaystyle m:=\inf\{k\in \mathbb {N} :x_{k}\neq y_{k}\}}$ ${\displaystyle (x_{k})=(y_{k})}$ ${\displaystyle d((x_{k}),(y_{x})):=0}$ ${\displaystyle \{0,\dotsc ,p-1\}}$
  • If , then the mapping is a homeomorphism of into the Cantor set . ${\displaystyle p=2}$ ${\displaystyle (x_{1},x_{2},\dotsc )\mapsto 2(x_{1}3^{-1}+x_{2}3^{-2}+x_{3}3^{-3}+\dotsb )}$ ${\displaystyle 2^{\mathbb {N} }}$
  • If a prime number, then the mapping is a homeomorphism from into the -adic integers. ${\displaystyle p}$ ${\displaystyle (x_{1},x_{2},\dotsc )\mapsto (x_{1}p^{0}+x_{2}p^{1}+x_{3}p^{2}+\dotsb )}$ ${\displaystyle p^{\mathbb {N} }}$ ${\displaystyle p}$
• The spectrum of any continuous linear operator on a Hilbert space is a compact subset of the complex numbers .
• The spectrum of any commutative ring or a Boolean algebra is a compact space with the Zariski topology .
• Further examples of compact sets from functional analysis can be obtained from the Banach-Alaoglu theorem , the Kolmogorow-Riesz theorem , the Arzelà-Ascoli theorem or the compactness criterion from James .

Not compact spaces

• The real numbers provided with the standard topology are not compact. The half-open interval , the whole numbers or the natural numbers are also not compact when viewed as subsets of . However, if one provides the trivial topology , for example , then is compact. Whether a set is compact therefore generally depends on the topology chosen. ${\displaystyle \mathbb {R} }$ ${\displaystyle [0,1[}$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle \mathbb {N} }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {N} }$ ${\displaystyle {\mathcal {T}}:=\{\emptyset ,\mathbb {N} \}}$ ${\displaystyle (\mathbb {N} ,{\mathcal {T}})}$
• The closed unit sphere of the space of bounded real number sequences (see L ^p -space ) is not compact, although it is closed and bounded. It is generally true that the closed unit sphere in a normalized space is compact if and only if the dimension of the space is finite. ${\displaystyle \ell ^{\infty }=L^{\infty }(\mathbb {N} ;\mathbb {R} )}$

properties

• The image of a compact set under a continuous function is compact. Hence a real-valued continuous function on a non-empty compact language takes on a global minimum and a global maximum .
• A continuous function on a compact metric space is uniformly continuous . This statement is also known as the Heine theorem.
• Every environment of a compact in a uniform space is a uniform environment . That is, it is with a neighborhood in the area. In the metric case, this means that all points with spheres of the same size of a selected size lie within the vicinity. The neighborhood can even be chosen in such a way that the complement of the surroundings with the neighborhood is outside of the compact with the neighborhood.
• Every infinite sequence of elements of a compact set has an accumulation point in . If the first axiom of countability is satisfied , then there is even a partial sequence that converges . However, the converse does not apply in every topological space, i.e. a subset in which every sequence has a convergent subsequence (in the subset) (such a subset is called sequence-compact, see below), does not have to be compact. (An example is the set of countable ordinal numbers with the order topology.) ${\displaystyle (a_{n})_{n\in \mathbb {N} }}$ ${\displaystyle K\subset E}$ ${\displaystyle K}$ ${\displaystyle K}$ ${\displaystyle K}$ ${\displaystyle (a_{n_{i}})_{i\in \mathbb {N} }}$
  ${\displaystyle [0,\omega _{1}[}$
• A closed subset of a compact space is compact.
• A compact subset of a Hausdorff room is complete.
• A non-empty compact subset of the real numbers has a largest and a smallest element (see also Supremum , Infimum ).
• For every subset of Euclidean space , the following three statements are equivalent (compare Heine-Borel's theorem ): ${\displaystyle M}$ ${\displaystyle \mathbb {R} ^{n}}$
  • ${\displaystyle M}$is compact, i.e. every open cover of has a finite partial cover . ${\displaystyle M}$
  • Each sequence in the set has a convergent subsequence (i.e. at least one accumulation point). ${\displaystyle M}$ ${\displaystyle M}$
  • The set is closed and limited . ${\displaystyle M}$
• A metric space is compact if and only if it is completely and totally restricted .
• The product of any class of compact spaces is compact in product topology . ( Tychonoff's theorem - this is equivalent to the axiom of choice )
• A compact Hausdorff room is normal .
• Every continuous bijective mapping from a compact space to a Hausdorff space is a homeomorphism .
• A metric space is compact if and only if every sequence in the space has a convergent subsequence with its limit value in the space.
• A topological space is compact if and only if every network in the space has a subnetwork that has a limit value in the space.
• A topological space is compact if and only if every filter in the space has a convergent refinement.
• A topological space is compact if and only if every ultrafilter converges on the space .
• A topological space can be embedded in a compact Hausdorff space if and only 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 .) ${\displaystyle X}$ ${\displaystyle X}$
• A metrizable space is compact if and only if every too homeomorphic metric space is complete. ${\displaystyle X}$ ${\displaystyle X}$
• If the metric space is compact and there is an open cover of , then there exists a number such that every subset of with diameter is contained in an element of the cover. ( Lemma of Lebesgue ) ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle \delta >0}$ ${\displaystyle X}$ ${\displaystyle {}<\delta }$
• Every compact Hausdorff space allows exactly one uniform structure that induces the topology. The reverse 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 partial cover, then the space is compact. ( Alexander's sub-base theorem )
• Two compact Hausdorff spaces and are accurate then homeomorphic if their rings of continuous real-valued functions and isomorphic are. ${\displaystyle X_{1}}$ ${\displaystyle X_{2}}$ ${\displaystyle C(X_{1})}$ ${\displaystyle C(X_{2})}$

Other forms of compactness

There are some topological properties that are equivalent to compactness in metric spaces but not equivalent in general topological spaces:

• Sequence compact: Each sequence has a convergent subsequence.
• ω-bounded : Every countable subset is contained in a compact subset.
• Countably compact : Every countable open cover has a finite partial cover. (Or, equivalently, every infinite subset has a cluster point.) ${\displaystyle \omega }$
• Pseudo-compact: Every real-valued continuous function in space is restricted.
• Weakly countable compact: Every infinite subset has an accumulation point .
• Eberlein-compact : The space is homeomorphic to a weakly compact subset of a Banach space .

While these concepts are equivalent for metric spaces, in general the following relationships exist:

• Compact spaces are limited. ${\displaystyle \omega }$
• ${\displaystyle \omega }$- restricted spaces are countably compact.
• Sequentially compact spaces are countably compact.
• Countably compact spaces are pseudocompact and weakly countably compact.
• Eberlein-compact rooms are consequently compact.

See also

• Relatively compact subset (compactness of the degree)
• Compactification (embedding topological spaces in compact spaces)
• Weakly follow compact set
• Complete algebraic variety (analogous concept formation for algebraic varieties)

literature

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

Web links

• Script on set theoretical topology ( Memento from June 11, 2007 in the Internet Archive ) (PDF; 1.72 MB).

Individual evidence

1. ↑ ^{a b c d} Boto von Querenburg : Set theoretical topology. 3rd, revised and expanded edition. Springer, Berlin et al. 2001, ISBN 3-540-67790-9
2. ↑ Winfried Kaballo : Basic Course Functional Analysis . 1st edition. Spektrum Akademischer Verlag, Heidelberg 2011, ISBN 978-3-8274-2149-4 , pp. 26 .
3. ^ Nicolas Bourbaki : Elements of Mathematics. General Topology. Volume 1. Springer, Berlin et al. 1966, Chapter II, § 4.3, Proposition 4.
4. ↑ István Sándor Gál : Uniformizable Spaces with a Unique Structure . In: Pacific Journal of Mathematics . tape 9 , no. 4 , August 1959, ISSN  0030-8730 , p. 1053-1060 ( online [PDF; 1,2 MB ]).