Compact Space

In mathematics, specifically general topology and metric topology, a compact space is a mathematical space in which any infinite sequence of points sampled from the space must eventually get arbitrarily close to some point of the space. There are several different notions of compactness, noted below, that are equivalent in good cases. The version just described is known as sequential compactness. The Bolzano–Weierstrass theorem gives an equivalent condition for sequential compactness when considering subsets of Euclidean space: a set then is compact if and only if it is closed and bounded. Examples include a closed interval or a rectangle. Thus if one chooses an infinite number of points in the closed unit interval, some of those points must get arbitrarily close to some real number in that space. For instance, some of the numbers 1/2, 4/5, 1/3, 5/6, 1/4, 6/7, … get arbitrarily close to 0. (Also, some get arbitrarily close to 1.) Note that the same set of points would not have, as an accumulation point, any point of the open unit interval; hence that space cannot be compact. Euclidean space itself is not compact since it is not bounded. In particular, no subset of the points 1, 2, 3, … on the real line gets arbitrarily close to any real number.

Apart from closed and bounded subsets of Euclidean space, typical examples of compact spaces include spaces consisting not of geometrical points but of functions. The term compact was introduced into mathematics by Maurice Fréchet in 1906 as a distillation of this concept. Compactness in this more general situation plays an extremely important role in mathematical analysis, because many classical and important theorems of 19th century analysis, such as the extreme value theorem, are easily generalized to this situation. A typical application is furnished by the Arzelà–Ascoli theorem and in particular the Peano existence theorem, in which one is able to conclude the existence of a function with some required properties as a limiting case of some more elementary construction.

Various equivalent notions of compactness, including sequential compactness and limit point compactness, can be developed in general metric spaces. In general topological spaces, however, the different notions of compactness are not necessarily equivalent, and the most useful notion, introduced by Pavel Alexandrov and Pavel Urysohn in 1929, involves the existence of certain finite families of open sets that "cover" the space in the sense that each point of the space must lie in some set contained in the family. This more subtle definition exhibits compact spaces as generalizations of finite sets. In spaces that are compact in this latter sense, it is often possible to patch together information that holds locally—that is, in a neighborhood of each point—into corresponding statements that hold throughout the space, and many theorems are of this character.