Compact Space - Theorems

Theorems

Some theorems related to compactness (see the glossary of topology for the definitions):

• A continuous image of a compact space is compact.
• The pre-image of a compact space under a proper map is compact.
• The extreme value theorem: a continuous real-valued function on a nonempty compact space is bounded above and attains its supremum. (Slightly more generally, this is true for an upper semicontinuous function.)
• A closed subset of a compact space is compact.
• A finite union of compact sets is compact.
• A nonempty compact subset of the real numbers has a greatest element and a least element.
• The product of any collection of compact spaces is compact. (Tychonoff's theorem, which is equivalent to the axiom of choice)
• Every topological space X is a dense subspace of a compact space having at most one point more than X, by the Alexandroff one-point compactification. By the same construction, every locally compact Hausdorff space X is a dense subspace of a compact Hausdorff space having at most one point more than X..
• Let X be a simply ordered set endowed with the order topology. Then X is compact if and only if X is a complete lattice (i.e. all subsets have suprema and infima).

Characterizations of compactness

(Assuming the axiom of choice), the following are equivalent.

1. A topological space X is compact.
2. Any collection of closed subsets of X with the finite intersection property has nonempty intersection.
3. X has a sub-base such that every cover of the space by members of the sub-base has a finite subcover (Alexander's sub-base theorem)
4. Every net on X has a convergent subnet (see the article on nets for a proof).
5. Every filter on X has a convergent refinement.
6. Every ultrafilter on X converges to at least one point.
7. Every infinite subset of X has a complete accumulation point.

Euclidean space

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

1. A is compact.
2. Every open cover of A has a finite subcover.
3. Every sequence in A has a convergent subsequence, whose limit lies in A.
4. Every infinite subset of A has at least one limit point in A.
5. A is closed and bounded (Heine–Borel theorem).
6. A is complete and totally bounded.

In practice, the condition (5) is easiest to verify, for example a closed interval or closed n-ball. Note that, in a metric space, every compact subset is closed and bounded. However, the converse may fail in non-Euclidean Rn. For example, the real line equipped with the discrete topology is closed and bounded but not compact, as the collection of all singleton points of the space is an open cover which admits no finite subcover.

Metric spaces

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

Hausdorff spaces

• A compact subset of a Hausdorff space is closed. More generally, compact sets can be separated by open sets: if K₁ and K₂ are compact and disjoint, there exist disjoint open sets U₁ and U₂ such that and . This is to say, compact Hausdorff space is normal.
• 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) Properties of the Banach space of continuous functions on a compact Hausdorff space are central to abstract analysis.
• Every continuous map from a compact space to a Hausdorff space is closed and proper (i.e., the pre-image of a compact set is compact.) In particular, every continuous bijective map from a compact space to a Hausdorff space is a homeomorphism.
• A topological space can be embedded in a compact Hausdorff space if and only if it is a Tychonoff space.