Compact space : Compactness 

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.

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