Compact space

Definition

A compact space is a concept in topological space which essentially states that, from a topological viewpoint, the space is small or limited. The term 'compact' was introduced by Maurice Fréchet in 1904 as a distillation of Bolzano-Weierstrass property in the real numbers, which says that every bounded sequence has a convergent subsequence. The concept of compactness is a generalization of this property to arbitrary topological spaces.

Formal Definition

In formal terms, a topological space is called compact if each of its open covers has a finite subcover. More specifically, given a space X, it is compact if for every collection {Uα} of open sets in X such that X is a subset of the union of the Uα, there exists a finite subcollection {U1, U2, ..., Un} such that X is a subset of the union of the Ui.

Properties

Compact spaces have several important properties that make them particularly interesting in the study of topology. These properties often make compact spaces easier to work with than non-compact spaces.

Finite Intersection Property

A collection of subsets of a topological space has the finite intersection property if the intersection of any finite subcollection is nonempty. A space is compact if and only if every collection of closed subsets with the finite intersection property has a nonempty intersection.

Limit Point Compactness

A space is limit point compact if every infinite subset has a limit point. For T1 spaces, limit point compactness, sequential compactness, and countable compactness are all equivalent to compactness.

Sequential Compactness

A space is sequentially compact if every sequence has a convergent subsequence. For metric spaces, sequential compactness is equivalent to compactness.

Countable Compactness

A space is countably compact if every countable open cover has a finite subcover. For first-countable spaces, countable compactness is equivalent to compactness.

Compactness in Various Contexts

Compactness can be defined in various mathematical contexts, each with its own nuances and implications.

Metric Spaces

In the context of metric spaces, a space is compact if it is complete and totally bounded. This is known as the Heine-Borel theorem.

Hausdorff Spaces

In Hausdorff spaces, every compact subset is closed. Furthermore, a compact Hausdorff space is a normal space, which is a space where every two disjoint closed sets can be separated by neighbourhoods.

Topological Groups

In the context of topological groups, a topological group is compact if and only if the underlying space is compact.

Compact Operators

In the context of operator theory, an operator is compact if it maps bounded sets to relatively compact sets.

Compactness and Continuity

Compactness has a deep connection with the concept of continuity. In particular, continuous functions preserve compactness, which is a key result in topology known as the Heine-Cantelli theorem.

Compactification

The process of embedding a topological space into a compact space in such a way that the original space is a dense subspace of the compact space is known as compactification. The most common form of compactification is the one-point compactification, also known as the Alexandroff compactification.

Compactness in Functional Analysis

In functional analysis, compactness plays a crucial role. The spectrum of a bounded operator in a Banach space is always a non-empty compact subset of the complex plane. Furthermore, the Riesz representation theorem states that the dual of a normed space is a Banach space.