In mathematics, a topological space is called compact if each of its open covers has a finite subcover. Otherwise it is called non-compact.

Note: Some authors such as Bourbaki use the term "quasi-compact" for this instead, and reserve the term "compact" for topological spaces that are both Hausdorff and "quasi-compact".

The Heine–Borel theorem shows that this definition is equivalent to "closed and bounded" for subsets of Euclidean space. So a subset of Euclidean space Rⁿ is called compact if it is closed and bounded. For example, in R, the closed unit interval 1 is compact, but the set of integers Z is not (it is not bounded) and neither is the half-open interval [0, 1) (it is not closed).

The concept of a compact subset of the real numbers can be extended to compact subsets of any topological space and indeed to the concept of a compact space. A subset is compact if when endowed with the subspace topology it becomes a compact space.

A single compact set is sometimes referred to as a compactum; following the Latin second declension (neuter), the corresponding plural form is compacta.

History and motivation

The identity of bounded closed subsets of real numbers and sets whose open covers have finite subcovers was discovered and proved in the late 19th century. See Heine–Borel theorem.

The term compact was introduced by Fréchet in 1906.

It has long been recognized that a property like compactness is necessary to prove many useful theorems. It used to be that "compact" meant "sequentially compact" (every sequence has a convergent subsequence). This was when primarily metric spaces were studied. The "covering compact" definition has become more prominent because it allows us to consider general topological spaces, and many of the old results about metric spaces can be generalized to this setting. This generalization is particularly useful in the study of function spaces, many of which are not metric spaces.

One of the main reasons for studying compact spaces is because they are in some ways very similar to finite sets: there are many results which are easy to show for finite sets, whose proofs carry over with minimal change to compact spaces. It is often said that "compactness is the next best thing to finiteness". Here is an example:

• Suppose X is a Hausdorff space, and we have a point x in X and a finite subset A of X not containing x. Then we can separate x and A by neighbourhoods: for each a in A, let U(x) and V(a) be disjoint neighbourhoods containing x and a, respectively. Then the intersection of all the U(x) and the union of all the V(a) are the required neighbourhoods of x and A.