Topological Completeness

Completeness is a property that can be defined on any metric space. A sequence of elements $(x_n)$ in a metric space $X$ is called a Cauchy Sequence iff given any $epsilon > 0$, there is a positive integer K such that forall elements in the sequence after K are epsilon-near. This means that forall $a > K$ and $b > K$, then $\delta( x_a, x_b ) < epsilon$.

A set is complete iff every Cauchy sequence in the space X converges to a point inside X. A subset, B of X is complete iff every Cauchy sequence within B converges to a point within B.

Be warned that some topological homeomorphisms can change whether a space is complete or not, so completeness is not a property purely of the topology of a space.

Analytic Completeness

The following statements are equivalent definitions of completeness in an analytic sense (perhaps only in the reals?):

• Every nonempty subset of the Reals which is bounded from above has a least upper bound in the Reals.
• Every nonempty subset of the Reals which is bounded from below has a greatest lower bound in the Reals.
• Every infinite bounded subset of Reals has a limit point.
• Every bounded sequence in the Reals has a convergent subsequence.