Complete space

2020 Mathematics Subject Classification: Primary: 54-XX [MSN][ZBL]

A term of relevance for a metric space, a uniform space, a topological space, a proximity space, the space of a topological group, a space with a symmetry, and a pseudo-metric space; it is also possible to use this term in still other cases. All definitions of completeness are based on a single general idea, the concrete expression of which depends on the particular type of space. The general feature in the various definitions of completeness lies in the requirement of convergence of a sufficiently broad class of sequences, directions or centred systems.

A metric space is called complete if each Cauchy sequence in it converges. In the same sense one understands the completeness of a pseudo-metric space and a space with a symmetry. A uniform space is called complete if for each centred system of sets in it containing sets which are arbitrarily small in relation to the coverings from the given uniform structure, the intersection of the elements of this system is not empty. On a topological group there are natural right- and left-uniform structures. If the space of the group in one of these structures is complete, then it is complete also in the other, and the topological group is then called Weyl complete. The term Raikov completeness is used in relation to the two-sided uniform structure on a group obtained by the union of the right- and left-uniform structures. Completeness of a metric space and Raikov completeness can be interpreted as absolute closure with respect to any representation of the given space as a subspace of a space of the same type. In particular, a metric space is complete if and only if it is closed in any metric space containing it. A topological group is Raikov complete if and only if it is closed in any topological group containing it as a topological subgroup. This is related to the fundamental construction of completions: Each metric space is put into correspondence with its completion in a canonical fashion, this being a complete metric space containing the original space as an everywhere-dense subspace. Similarly, each topological group is Raikov completeable, but not every topological group is Weyl completeable.

For topological spaces, the requirement of absolute closure (i.e. closure in any space containing it) leads to compact spaces if one restricts oneself to the class of completely-regular Hausdorff spaces: Those spaces and only those spaces have this property. However, there is another useful and natural approach to defining completeness in a topological space. A completely-regular Hausdorff space is called Čech complete if it can be represented as the intersection of a countable family of open sets in a certain Hausdorff compactification. All such spaces have the Baire property: The intersection of a countable family of non-empty open everywhere-dense sets is always non-empty. A metrizable space is Čech complete if and only if it is metrizable by a complete metric (the Aleksandrov–Hausdorff theorem). Čech completeness provides for correct behaviour of topological spaces in many important respects. For example, a countable Čech-complete space has a countable base and is metrizable. Paracompactness is retained in the product operation when the spaces are Čech complete. Čech completeness is also preserved by perfect mappings (cf. Perfect mapping), and in the class of metrizable spaces it is preserved under transformation by open continuous mappings.

Another useful approach to defining completeness in a completely-regular Hausdorff space is related to considering the maximal uniform structure on it: If such a uniform space is complete, the topological space is called Dieudonné complete. Dieudonné completeness applies to precisely those spaces that are homeomorphic to closed subspaces of a topological product of metrizable spaces. In the presence of Dieudonné completeness, a single property represents pseudo-compactness, countable compactness and compactness. All paracompact spaces are Dieudonné complete, and this applies in particular to all metric spaces. This shows that Dieudonné completeness does not imply that the Baire property holds in the space. A special case of Dieudonné completeness is Hewitt completeness of a topological space, which means that the space is homeomorphic to a closed subspace of the topological product of a certain family of real straight lines.

References

[1]	A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) (Translated from Russian)

Comments

A Hewitt-complete space is also called a realcompact space. A centred system of sets is a collection of sets such that finite intersections are non-empty.