Compact space

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 use the term 'quasicompact' instead and reserve the term 'compact' for compact Hausdorff spaces, but Wikipedia follows the usual current practice of allowing compact spaces to be non-Hausdorff.

Equivalent definitions of a compact set in Rⁿ

For any subset of Euclidean space Rⁿ, the following three conditions are equivalent:

• Every open cover has a finite subcover. This is the definiton most commonly used, as stated above.
• Every sequence in the set has a convergent subsequence.
• The set is closed and bounded. This is the condition that is easiest to verify, for example a closed interval or closed n-ball.

In other spaces, these conditions may or may not be equivalent, depending on the properties of the space.

Examples of compact spaces

• The closed unit interval [0,1] (but not the half-open interval [0,1)).
• The n-sphere, for every natural number n.
• The Cantor set (and the p-adic integers, which are homeomorphic to the Cantor set).
• Any finite topological space.
• Any space carrying the cofinite topology[?] (a set being open iff it is empty or its complement is finite).
• Consider the set 2^N of all infinite sequences with entries in {0,1}. We can turn it into a metric space by defining d((x_n),(y_n)) = 1/k, where k is the smallest index such that x_k ≠ y_k (if there is no such index, then the two sequences are the same, and we define their distance to be zero). Then 2^N is a compact space, a consequence of Tychonoff's theorem mentioned below. This construction can be performed for any finite set, not just {0,1}.
• The spectrum of any continuous linear operator on a Hilbert space is a compact subset of C.
• The spectrum of any commutative ring or Boolean algebra.
• The Hilbert cube.
• The right order topology[?] or left order topology[?] on any bounded totally ordered set, in particular Sierpinski space[?].
• The Stone-Čech compactification of any Tychonoff space is a compact Hausdorff space.
• The Alexandroff one-point compactification of any space is compact.

Theorems

Some theorems related to compactness (see the Topology Glossary for the definitions):

• A continuous image of a compact space is compact.

• 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 is closed and bounded. (Heine-Borel theorem)

• A metric space (or uniform space) is compact if and only if it is complete and totally bounded.

• The product of any collection of compact spaces is compact. (Tychonoff's theorem -- this is equivalent to the axiom of choice)

• A compact Hausdorff space is normal.

• Every continuous bijective map from a compact space to a Hausdorff space is a homeomorphism.

• A metric space is compact if and only if every sequence in the space has a subsequence with limit in the space.

• A topological space is compact if and only if every net on the space has a subnet which has a limit in the space.

• 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 is a Tychonoff space.

• Every topological space X is a dense subspace of a compact space which has at most one point more than X. (Alexandroff one-point compactification)

• A metric space X is compact if and only if every metric space homeomorphic to X is complete.

• 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.

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: Every infinite subset has an accumulation point.

While all these concepts are equivalent for metric spaces, in general we have the following implications:

Compact spaces are countably compact. Sequentially compact spaces are countably compact. Countably compact spaces are pseudocompact and weakly countably compact.