### 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.