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Compact space. |
In mathematics, a subset of Euclidean space R^{n} is called compact if it is closed and bounded. For example, in R, the closed unit interval [0, 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). A more modern approach is to call a topological space compact if each of its open covers has a finite subcover. The Heine-Borel theorem shows that this definition is equivalent to "closed and bounded" for subsets of Euclidean space. Note: Some authors such as Bourbaki use the term "quasi-compact" instead and reserve the name "compact" for topological spaces that are Hausdorff and compact. History and motivation of compact space. The term compact was introduced by Fréchet in 1906. It has long been recognized that a property like compactness is necessary to prove a lot of 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. 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:
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 axiom. Many of the arguments and results involving compact spaces follow such a pattern. Definitions of compact space. Compactness of subsets of R^{n} For any subset of Euclidean space R^{n}, the following four conditions are equivalent:
In other spaces, these conditions may or may not be equivalent, depending on the properties of the space. Compactness of topological spaces of compact space. The "finite subcover" property from the previous paragraph is more abstract than the "closed and bounded" one, but it has the distinct advantage that it can be given using the subspace topology on a subset of R^{n}, eliminating the need of using a metric or an ambient space. Thus, compactness is a topological property. In a sense, the closed unit interval [0,1] is intrinsically compact, regardless of how it is embedded in R or R^{n}. A topological space X is defined as compact if all its open covers have a finite subcover. Formally, this means that
An often used equivalent definition is given in terms of the finite intersection property: if any collection of closed sets satisfying the finite intersection property has nonempty intersection, then the space is compact. This definition is dual to the usual one stated in terms of open sets. Some authors require that a compact space also be Hausdorff, and the non-Hausdorff version is then called quasicompact. Examples of compact spaces.
Theorems of compact space. Some theorems related to compactness (see the Topology Glossary for the definitions):
Other forms of compact space. There are a number of topological properties which are equivalent to compactness in metric spaces, but are inequivalent in general topological spaces. These include the following.
While all these conditions are equivalent for metric spaces, in general we have the following implications:
Not every countably compact space is compact; an example is given by the first uncountable ordinal with the order topology. Not every compact space is sequentially compact; an example is the infinite product space 2^{[0, 1]} with the product topology. A metric space is called pre-compact or totally bounded if any sequence has a Cauchy subsequence; this can be generalised to uniform spaces. For complete metric spaces this is equivalent to compactness. See relatively compact for the topological version. Another related notion that is strictly weaker than compactness is local compactness. Why not also search for...
References to compact space.
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