Some Theorems
A metric space X is complete if and only if every decreasing sequence of non-empty closed subsets of X, with diameters tending to 0, has a non-empty intersection: if Fn is closed and non-empty, Fn + 1 ⊂ Fn for every n, and diam(Fn) → 0, then there is a point x ∈ X common to all sets Fn.
Every compact metric space is complete, though complete spaces need not be compact. 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 S of Rn is compact and therefore complete.
A closed subspace of a complete space is complete. Conversely, a complete subset of a metric space is closed.
If X is a set and M is a complete metric space, then the set B(X, M) of all bounded functions ƒ from X to M is a complete metric space. Here we define the distance in B(X, M) in terms of the distance in M with the supremum norm
If X is a topological space and M is a complete metric space, then the set Cb(X, M) consisting of all continuous bounded functions ƒ from X to M is a closed subspace of B(X, M) and hence also complete.
The Baire category theorem says that every complete metric space is a Baire space. 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. The fixed point theorem is often used to prove the inverse function theorem on complete metric spaces such as Banach spaces.
The expansion constant of a metric space is the infimum of all constants such that whenever the family intersects pairwise, the intersection
is nonempty. A metric space is complete if and only if its expansion constant is ≤ 2.
Read more about this topic: Complete Metric Space