Everything about Compact Set totally explained
In
mathematics, a
subset of
Euclidean space Rn is called
compact if it's
closed and
bounded. For example, in
R, the closed
unit interval [0,1] is compact, but the set of
integers
Z isn't (it isn't bounded) and neither is the half-open interval
[0,1) (it isn't 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 term "compact" for topological spaces that are
Hausdorff and "quasi-compact".
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 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're 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've 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 ^2 is homeomorphic to the sphere
. Using the one-point compactification, one can also easily construct compact spaces which are not Hausdorff, by starting with a non-Hausdorff space.
The spectrum of any continuous linear operator on a Hilbert space is a compact subset of the complex numbers C. If the Hilbert space is infinite-dimensional, then any compact subset of C arises in this manner, as the spectrum of some continuous linear operator on the Hilbert space.
The spectrum of any commutative ring or Boolean algebra is compact.
The Hilbert cube is compact.
The right order topology or left order topology on any bounded totally ordered set is compact. In particular, Sierpinski space is compact.
The prime spectrum of any commutative ring with the Zariski topology is a compact space, important in algebraic geometry. These prime spectra are almost never Hausdorff spaces.
Theorems
Some theorems related to compactness (see the Topology Glossary for the definitions):
A continuous image of a compact space is compact.
The extreme value theorem: a continuous real-valued function on a nonempty compact space is bounded and attains its supremum.
A closed subset of a compact space is compact.
A compact subset of a Hausdorff space is closed.
A nonempty compact subset of the real numbers has a greatest element and a least element.
A subset of Euclidean n-space is compact if and only if it's closed and bounded. (Heine–Borel theorem)
A metric space (or uniform space) is compact if and only if it's complete and totally bounded.
The product of any collection of compact spaces is compact. (Tychonoff's theorem, which is equivalent to the axiom of choice)
A compact Hausdorff space is normal.
Every continuous map from a compact space to a Hausdorff space is closed and proper. It follows that every continuous bijective map from a compact space to a Hausdorff space is a homeomorphism.
A metric space (or more generally any first-countable uniform space) is compact if and only if every sequence in the space has a convergent subsequence.
A topological space is compact if and only if every net on the space has a convergent subnet.
A topological space is compact if and only if every filter on the space has a convergent refinement.
A topological space is compact if and only if every ultrafilter on the space is convergent.
A topological space can be embedded in a compact Hausdorff space if and only if it's a Tychonoff space.
Every non-compact topological space X is a dense subspace of a compact space which has at most one point more than X. (Alexandroff one-point compactification)
If the metric space X is compact and an open cover of X is given, then there exists a number δ > 0 such that every subset of X of diameter < δ is contained in some member of the cover. (Lebesgue's number lemma)
If a topological space has a sub-base such that every cover of the space by members of the sub-base has a finite subcover, then the space is compact. (Alexander's sub-base theorem)
Two compact Hausdorff spaces X1 and X2 are homeomorphic if and only if their rings of continuous real-valued functions C(X1) and C(X2) are isomorphic. (Gelfand-Naimark theorem)
Other forms of compactness
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.
Sequentially compact: Every sequence has a convergent subsequence.
Countably compact: Every countable open cover has a finite subcover. (Or, equivalently, every infinite subset has an ω-accumulation point.)
Pseudocompact : Every real-valued continuous function on the space is bounded.
Weakly countably compact (or limit point compact): Every infinite subset has an accumulation point.
While all these conditions are equivalent for metric spaces, in general we've the following implications:
Compact spaces are countably compact.
Sequentially compact spaces are countably compact.
Countably compact spaces are pseudocompact and weakly countably compact.
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 given by 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 which (by most definitions) is strictly weaker than compactness is local compactness.
Further Information
Get more info on 'Compact Set'.
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