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In mathematics, a compact space is a space that resembles a closed and bounded subset of Euclidean space Rn in that it is "small" in a certain sense and "contains all its limit points". The modern general definition calls a topological space compact if every open cover of it has a finite subcover. That is, any collection of open sets whose union is the whole space has a finite subcollection whose union is still the whole space. Some authors (including Bourbaki) use the term quasicompact instead and reserve the term compact for compact Hausdorff spaces, but this encyclopedia follows the usual current practice of allowing compact spaces to be non-Hausdorff.1 Motivation for compactness
One of the main reasons for studying compact spaces is because they are in some ways very similar to finite sets. In other words, there are many results which are easy to show for finite sets, the proofs of which 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.
Note that if A is infinite, the proof fails, because the intersection of arbitrarily many neighbourhoods of x might not be a neighbourhood of x. The proof can be "rescued", however, if A is compact: we simply take a finite subcover of the cover {V(a)} of A. In this way, we see that in a Hausdorff space, any point can be separated by neighbourhoods from any compact set not containing it. In fact, repeating the argument shows that any two disjoint compact sets in a Hausdorff space can be separated by neighbourhoods -- note that this is precisely what we get if we replace "point" (i.e. singleton set) with "compact set" in the Hausdorff separation axiomIn topology and related fields of mathematics, there are several restrictions that one often makes on the kinds of topological spaces that one wishes to consider. Some of these restrictions are given by the separation axioms . These are sometimes called T. Many of the arguments and results involving compact spaces follow such a pattern.
2 Generally equivalent definitions of compact sets
An equivalent definition of compact spaces, sometimes useful, is based on the finite intersection propertyIn topology, the finite intersection property is a property of a collection of subsets of a set X. A collection has this property if the intersection over any finite subcollection of the collection is nonempty. This is trivially satisfied if the intersect. This definition says that X is compact if and only if for every collection of closed sets which has the finite intersection property, the intersection over this collection is also nonempty. In other words, if all finite subsets of a collection of closed sets have nonempty intersection, so must the entire collection. For example, (0, 1] is not compact, since the sequence (0, 1/n] of closed sets (in (0, 1]) is nested, and so clearly has the finite intersection property, but has empty intersection. This definition is used in some proofs of Tychonoff's theoremIn mathematics, Tychonoff's theorem states that the product of any collection of compact topological spaces is compact. For finite collections of compact spaces, this is not very surprising. The statement is in fact true for infinite collections of arbitr and
the uncountabilityIn mathematics, an uncountable set is a set which is not countable. Here, "countable" means countably infinite or finite, so by definition, all uncountable sets are infinite. The best known example of an uncountable set is the set R of all real numbers; C of the real numberIn mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. The term "real number" is a retronym coined in response to " imaginary number". Real numbers mays.
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