Completeness is fundamentally a metric property
(the definition of completeness depends critically on the definition of a distance).
Even if two distances are defined on the same set which induce the
same topology on that space, it's quite possible that one distance
defines a complete space and the other one doesn't.

A metric space is
compact if and only if it's
complete and totally bounded.

A subset of a metric space is said to be totally bounded when it can be covered by finitely
many balls of radius r, for any given radius r.
In a Euclidean space of infinitely many dimensions, a bounded set
(like a ball of unit radius) need not be totally bounded.
Actually, a closed ball is compact only in a space of finitely many dimensions.

By definition, a topological property is preserved
by any homeomorphism.
This is not always the case for completeness.
For example, is complete and it's homeomorphic to
the open interval ]0,1[
which is not.
(HINT : A positive sequence that tends to
0 in [0,1] isn't convergent in ]0,1[ .)

A metrizable space is defined as a topological space
homeomorphic to a metric space.
Such a space is called complete-metrizable when at least one
metric space homeomorphic to it is complete.
That's a topological property (since it's clearly preserved by homeomorphisms)
but it's difficult to characterize in practice.

(2014-12-05) Completeness in a Uniform Space
(Weil, 1937)
Completeness can also be defined in uniform topological spaces.

A uniform space is complete
when every Cauchy filter in it converges.

Motivation :

What made it possible to define completeness in a metric space is the existence of a
family of relations (i.e., subsets of the cartesian product)
dependent on a single positive parameter a:

U_{a} = { (x,y) | d(x,y) < a }

The triangular inequality for the distance d enables us to
construct a relation V = U_{a/2 } which is. loosely speaking,
at most half as wide as the relation U = U_{a }.
The crucial aspect can be expressed as follows, in terms of
the composition of relations
(this simple exercise is left to the reader).

V o V Í U

This expression no longer involves any explicit reference to distances. The postulated existence
of a sequence of relations based on this composition pattern will enable us to generalize
the notion of Cauchy sequences and completeness without using the notion of a distance...

For that reason, Henri Garnir (1921-1985)
has proposed to adopt DC or, equivalently, the Baire Category theorem instead of AC
among the axioms of Set theory.
This allowed him to postulate that every set of reals is
almost open,
which makes every set of reals Lebesgue-measurable and dismisses the
Banach-Tarski paradox.

(2007-11-15) Completeness Redux
Tentative (flawed)
topological characterizations of completeness.

Let's try topological characterizations
of completeness to see how such attempts fail.
For example, let's examine the following property:

Any decreasing sequence of nonempty closed sets has a nonempty intersection:

{ " i
Î
,
A_{i}
¹ Æ is closed,
A_{i+1} Í A_{i}
}
Þ
Æ ¹

Ç

A_{i}

i Î

This would seem like a good candidate for a topological characterization of completeness
until you realize that it's not even true for a noncompact complete space like
in which there are indeed
nested collection of nonempty closed sets with an empty intersection.
One example is
A_{i} = [i,¥[.

For families of compact closed sets,
the above characterization still fails for metric spaces of infinitely
many dimensions (where closed balls are not compact).

All told, a topological space can only be said to be complete with respect to
a specific distance compatible with its topology
(two different distances may induce the same topology but the space can be complete
with respect to one metric and not the other).
Such a space is called either topologically complete
or complete-metrizable. There is simply no easy way to characterize
that property...

A Banach space is a
normed vector space which is complete
(which is to say that every Cauchy sequence
in it converges).
The concept is named after the Polish mathematician
Stefan Banach (1892-1945) who
axiomatized the idea in his doctoral dissertation (1920)
and made it popular through his 1931 foundational book on functional analysis, which
was translated in French the next year
(Théorie des opérations linéaires, 1932).

Arguably, Banach spaces are the main backdrop for modern analysis,
the branch of mathematics which revolves around the very notion of
limit
(it would be hazardous to discuss limits in a space that's not complete).

The key example which motivated Stefan Banach :

The Riesz-Fischer theorem (1907) states that
L^{p} is a Banach space.

(2013-01-22) Fréchet Spaces
The key properties of Banach spaces for distances not based on a norm.

In 1906, Maurice
Fréchet had proposed the general notion of a metric space
without assuming an underlying vector structure.

He realized that the key results that make
Banach spaces interesting could also be obtained
for vector spaces that are complete with respect to a distance not
associated with a norm. He thus investigated structures more general
than Banach spaces, which are now called Fréchet spaces :

A Fréchet space is a locally-convex vector space
which is complete with respect to a given translation-invariant metric.