Compact space

[1] The idea is that a compact space has no "punctures" or "missing endpoints", i.e., it includes all limiting values of points.

[2] The Bolzano–Weierstrass theorem states that a subset of Euclidean space is compact in this sequential sense if and only if it is closed and bounded.

[3] In general topological spaces, however, these notions of compactness are not necessarily equivalent.

The most useful notion — and the standard definition of the unqualified term compactness — is phrased in terms of the existence of finite families of open sets that "cover" the space, in the sense that each point of the space lies in some set contained in the family.

In the 19th century, several disparate mathematical properties were understood that would later be seen as consequences of compactness.

The full significance of Bolzano's theorem, and its method of proof, would not emerge until almost 50 years later when it was rediscovered by Karl Weierstrass.

[4] In the 1880s, it became clear that results similar to the Bolzano–Weierstrass theorem could be formulated for spaces of functions rather than just numbers or geometrical points.

The idea of regarding functions as themselves points of a generalized space dates back to the investigations of Giulio Ascoli and Cesare Arzelà.

Towards the beginning of the twentieth century, results similar to that of Arzelà and Ascoli began to accumulate in the area of integral equations, as investigated by David Hilbert and Erhard Schmidt.

For a certain class of Green's functions coming from solutions of integral equations, Schmidt had shown that a property analogous to the Arzelà–Ascoli theorem held in the sense of mean convergence – or convergence in what would later be dubbed a Hilbert space.

However, a different notion of compactness altogether had also slowly emerged at the end of the 19th century from the study of the continuum, which was seen as fundamental for the rigorous formulation of analysis.

The significance of this lemma was recognized by Émile Borel (1895), and it was generalized to arbitrary collections of intervals by Pierre Cousin (1895) and Henri Lebesgue (1904).

The Heine–Borel theorem, as the result is now known, is another special property possessed by closed and bounded sets of real numbers.

This sentiment was expressed by Lebesgue (1904), who also exploited it in the development of the integral now bearing his name.

Ultimately, the Russian school of point-set topology, under the direction of Pavel Alexandrov and Pavel Urysohn, formulated Heine–Borel compactness in a way that could be applied to the modern notion of a topological space.

A nontrivial example of a compact space is the (closed) unit interval [0,1] of real numbers.

The given example sequence shows the importance of including the boundary points of the interval, since the limit points must be in the space itself — an open (or half-open) interval of the real numbers is not compact.

A subset of Euclidean space in particular is called compact if it is closed and bounded.

[3] In contrast, the different notions of compactness are not equivalent in general topological spaces, and the most useful notion of compactness – originally called bicompactness – is defined using covers consisting of open sets (see Open cover definition below).

That this form of compactness holds for closed and bounded subsets of Euclidean space is known as the Heine–Borel theorem.

Formally, a topological space X is called compact if every open cover of X has a finite subcover.

Some branches of mathematics such as algebraic geometry, typically influenced by the French school of Bourbaki, use the term quasi-compact for the general notion, and reserve the term compact for topological spaces that are both Hausdorff and quasi-compact.

[13] For any subset A of Euclidean space, A is compact if and only if it is closed and bounded; this is the Heine–Borel theorem.

In addition, the following are equivalent for all ordered spaces (X, <), and (assuming countable choice) are true whenever (X, <) is compact.

A topological space X is pseudocompact if and only if every maximal ideal in C(X) has residue field the real numbers.

For completely regular spaces, this is equivalent to every maximal ideal being the kernel of an evaluation homomorphism.

The framework of non-standard analysis allows for the following alternative characterization of compactness:[17] a topological space X is compact if and only if every point x of the natural extension *X is infinitely close to a point x0 of X (more precisely, x is contained in the monad of x0).

For example, an open real interval X = (0, 1) is not compact because its hyperreal extension *(0,1) contains infinitesimals, which are infinitely close to 0, which is not a point of X.