In mathematical analysis, a metric space M is called complete (or a Cauchy space) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if there are no "points missing" from it (inside or at the boundary).
For instance, the set of rational numbers is not complete, because e.g.
is "missing" from it, even though one can construct a Cauchy sequence of rational numbers that converges to it (see further examples below).
It is always possible to "fill all the holes", leading to the completion of a given space, as explained below.
is called Cauchy if for every positive real number
is complete if any of the following equivalent conditions are satisfied: The space
of rational numbers, with the standard metric given by the absolute value of the difference, is not complete.
The open interval (0,1), again with the absolute difference metric, is not complete either.
of complex numbers (with the metric given by the absolute difference) are complete, and so is Euclidean space
becomes a complete metric space if we define the distance between the sequences
In fact, a metric space is compact if and only if it is complete and totally bounded.
This is a generalization of the Heine–Borel theorem, which states that any closed and bounded subspace
That is, the union of countably many nowhere dense subsets of the space has empty interior.
The Banach fixed-point theorem states that a contraction mapping on a complete metric space admits a fixed point.
It has the following universal property: if N is any complete metric space and f is any uniformly continuous function from M to N, then there exists a unique uniformly continuous function f′ from M′ to N that extends f. The space M' is determined up to isometry by this property (among all complete metric spaces isometrically containing M), and is called the completion of M. The completion of M can be constructed as a set of equivalence classes of Cauchy sequences in M. For any two Cauchy sequences
(This limit exists because the real numbers are complete.)
This is only a pseudometric, not yet a metric, since two different Cauchy sequences may have the distance 0.
But "having distance 0" is an equivalence relation on the set of all Cauchy sequences, and the set of equivalence classes is a metric space, the completion of M. The original space is embedded in this space via the identification of an element x of M' with the equivalence class of sequences in M converging to x (i.e., the equivalence class containing the sequence with constant value x).
This defines an isometry onto a dense subspace, as required.
Notice, however, that this construction makes explicit use of the completeness of the real numbers, so completion of the rational numbers needs a slightly different treatment.
The additional subtlety to contend with is that it is not logically permissible to use the completeness of the real numbers in their own construction.
Nevertheless, equivalence classes of Cauchy sequences are defined as above, and the set of equivalence classes is easily shown to be a field that has the rational numbers as a subfield.
This field is complete, admits a natural total ordering, and is the unique totally ordered complete field (up to isomorphism).
The truncations of the decimal expansion give just one choice of Cauchy sequence in the relevant equivalence class.
In topology one considers completely metrizable spaces, spaces for which there exists at least one complete metric inducing the given topology.
Since the conclusion of the Baire category theorem is purely topological, it applies to these spaces as well.
Since Cauchy sequences can also be defined in general topological groups, an alternative to relying on a metric structure for defining completeness and constructing the completion of a space is to use a group structure.
This is most often seen in the context of topological vector spaces, but requires only the existence of a continuous "subtraction" operation.
A common generalisation of these definitions can be found in the context of a uniform space, where an entourage is a set of all pairs of points that are at no more than a particular "distance" from each other.
The most general situation in which Cauchy nets apply is Cauchy spaces; these too have a notion of completeness and completion just like uniform spaces.