Compact space

Compact space is a certain type of topological spaces that generalizes the properties of boundedness and closure in Euclidean spaces to arbitrary topological spaces.

In general topology, compact spaces in their properties resemble finite sets in set theory .

Content

History

A compact space is a concept introduced by Aleksandrov to strengthen the concept of compact space defined by Maurice Frechet : a topological space is compact - in the original sense of the word - if each countable open cover of this space contains its finite subcover ^[1] . However, the further development of mathematics showed that the concept of compactness is so much more important than the original concept of compactness, that compactness is understood as compactness at present, and compact spaces in the old sense are called countably compact . Both concepts are equivalent when applied to metric spaces.

Definition

A compact space is a topological space in any covering of which by open sets there is a finite subcovering .

Examples of compact sets

Closed bounded sets in $\mathbb {R} ^{n}$ ${\ displaystyle \ mathbb {R} ^ {n}}$ .
Finite subsets of topological spaces.
The Ascoli – Arzela theorem gives a characterization of compact sets for some function spaces. Consider space $C(X)$ ${\ displaystyle C (X)}$ real functions on a metric compact space $X$ ${\ displaystyle X}$ with the norm $\|f\|=\sup _{x}|f(x)|$ ${\ displaystyle \ | f \ | = \ sup _ {x} | f (x) |}$ . Then the closure of the set of functions $F$ ${\ displaystyle F}$ at $C(X)$ ${\ displaystyle C (X)}$ compact if and only if $F$ ${\ displaystyle F}$ uniformly bounded and equi-continuous .
Stone space of Boolean algebras.
Compactification of topological space.

Related Definitions

A subset of a topological space T , which is a compact space in the topology induced by T , is called a compact set .
A set is called precompact (or compact with respect to T ) if its closure in T is compact ^[2] .
A space is called sequentially compact if a convergent subsequence can be distinguished from any sequence in it.
A locally compact space is a topological space in which every point has a neighborhood whose closure is compact.
A boundedly compact space is a metric space in which all closed balls are compact.
A pseudocompact space is a Tikhonov space in which every continuous real function is bounded.
A countably compact space is a topological space in which any countable cover of which by open sets there is a finite subcover.
A weakly countably compact space is a topological space in which any infinite set has a limit point.
An H-closed space is a Hausdorff space closed in any enclosing space ^[3] ^[4] .

The term “ compact ” is sometimes used for a metrizable compact space, but sometimes simply as a synonym for the term “compact space”. Also, a “ compact ” is sometimes used for a Hausdorff compact space ^[5] . Next, we will use the term “ compact ” as a synonym for the term “compact space”.

Compact and quasi-compact

French word fr. compact is the translator’s false friend : it means not “compact,” but “compact and Hausdorff .” Compact spaces that are not a priori Hausdorff spaces are called French in French. quasi-compact . Such usage was introduced in Bourbaki treatises. In other languages, it is not generally accepted, with the exception of works on abstract algebraic geometry . The basic object of abstract algebraic geometry, the spectrum of the ring , always compact space, but almost never Hausdorff; due to the influence of the works of Grothendieck , who relied on Bourbaki, in texts on abstract algebraic geometry the word quasi-compact , generally inconsistent with tradition, is sometimes allowed ^[6] .

Properties

Properties equivalent to compactness:
- A topological space is compact if and only if every centered family of closed sets, that is, a family in which the intersections of finite subfamilies are not empty, has a nonempty intersection ^[7] .
- A topological space is compact if and only if every direction in it has a limit point.
- A topological space is compact if and only if every filter in it has a limit point.
- A topological space is compact if and only if each ultrafilter converges to at least one point.
- Topological space $X$ ${\ displaystyle X}$ compact if and only if in it every infinite subset has at least one point of complete accumulation in $X$ ${\ displaystyle X}$ .
Other common properties:
- For any continuous mapping, the image of a compact is a compact.
- Weierstrass theorem . Any continuous real function on a compact space is bounded and reaches its largest and smallest values.
- A closed subset of a compact set is compact.
- A compact subset of Hausdorff space is closed .
- A compact Hausdorff space is normal .
- A Hausdorff space is compact if and only if it is regular and H-closed ^[3] ^[4] .
- A Hausdorff space is compact if and only if each of its closed subsets is H-closed ^[3] ^[4] .
- Tikhonov's theorem: the product of an arbitrary (not necessarily finite) set of compact sets (with the product topology ) is compact.
- Any continuous one-to-one mapping of a compact set into a Hausdorff space is a homeomorphism .
- Compact sets “behave like points” ^[8] . For example: in a Hausdorff space, any two disjoint compact sets have disjoint neighborhoods, in a regular space any disjoint compact and closed sets have disjoint neighborhoods, in a Tikhonov space any disjoint compact and closed sets are functionally separable .
- Each finite topological space is compact.

Properties of compact metric spaces:
- A metric space is compact if and only if any sequence of points in it contains a convergent subsequence.
- The Hausdorff compactness theorem gives necessary and sufficient compactness conditions for a set in a metric space.
- For finite - dimensional Euclidean spaces, a subspace is compact if and only if it is bounded and closed . About spaces with such a property, they say that they satisfy the Heine – Borel property ^[9] .
- Lebesgue lemma : for any compact metric space and open cover $\{V_{\alpha }\},\ \alpha \in A$ ${\ displaystyle \ {V _ {\ alpha} \}, \ \ alpha \ in A}$ there is a positive number $r$ ${\ displaystyle r}$ such that any subset whose diameter is smaller $r$ ${\ displaystyle r}$ contained in one of the sets $V_{\alpha }$ ${\ displaystyle V _ {\ alpha}}$ . Such a number $r$ ${\ displaystyle r}$ called the Lebesgue number.

Notes

↑ Bicompact space, mathematical encyclopedia
↑ Kolmogorov A.N., Fomin S.V. Elements of the theory of functions and functional analysis. - 6th edition, revised. Moscow, Science, 1989 (p. 123)
↑ ¹ ² ³ Kelly, p. 209
↑ ¹ ² ³ H-closed space - an article from the mathematical encyclopedia . V.I. Ponomarev.
↑ Engelking, p. 208
↑ B.K. Zavyalov . Brauer Group and Field Extensions II. Question 1. , 2017
↑ See also the Lemma on nested segments.
↑ Engelking, p. 210
↑ See also Bolzano-Weierstrass Theorem # Bolzano-Weierstrass Theorem and the concept of compactness

Literature

Kolmogorov A.N., Fomin S.V. Elements of the theory of functions and functional analysis. - Any edition.
O. Ya. Viro, O.A. Ivanov, V.M. Kharlamov and N. Yu. Netsvetaev . Topology Tutorial
Protasov V. Yu. Maxima and minima in geometry . - M .: ICMMO. - 56 p. - (Library "Mathematical Education", issue 31).
L. Schwartz , Analysis, vol. I, M., Mir, 1972.
Engelking, R. General Topology. - M .: Mir , 1986 .-- 752 p.
Arkhangelsky A.V. Compact space // Mathematical Encyclopedia . - M .: Soviet Encyclopedia, 1977-1985.
Wojciechowski M.I. Compact space // Mathematical Encyclopedia . - M .: Soviet Encyclopedia, 1977-1985.

[1] Bicompact space, mathematical encyclopedia

[KF-2] Kolmogorov A.N., Fomin S.V. Elements of the theory of functions and functional analysis. - 6th edition, revised. Moscow, Science, 1989 (p. 123)

[:0-3] ¹ ² ³ Kelly, p. 209

[:1-4] ¹ ² ³ H-closed space - an article from the mathematical encyclopedia . V.I. Ponomarev.

[5] Engelking, p. 208

[6] B.K. Zavyalov . Brauer Group and Field Extensions II. Question 1. , 2017

[7] See also the Lemma on nested segments.

[8] Engelking, p. 210

[9] See also Bolzano-Weierstrass Theorem # Bolzano-Weierstrass Theorem and the concept of compactness