A space (in the sense of topology) is complete (or Cauchy complete) if every sequence, net, or filter that should converge really does converge. We identify the sequences, nets, or filters that “should” converge as the Cauchy ones.
A space that is not complete has “gaps” that may be filled to form its completion; it is rather natural to make the space Hausdorff at the same time. Forming the completion of a Hausdorff space is an important example of completion in the general sense.
A space (which may be a metric space, a Cauchy space, or anything in between) is Cauchy complete if every Cauchy filter converges, equivalently if every Cauchy net converges. A space is sequentially complete if every Cauchy sequence converges. Note that a sequentially complete metric space must be complete, but this does not hold for more general spaces.
A space with an underlying topology is topologically complete if the underlying topological space is completely metrizable?; that is with the property of having some complete metric on its underlying set that is compatible with the underlying topology. In other words, the groupoid of topologically complete spaces is (up to equivalence) the groupoid of completely metrizable topological spaces, or equivalently the category whose objects are complete metric spaces and whose morphisms are homeomorphisms. This is, of course, a terrible name, in part because it may be confused with the more general Dieudonné-complete spaces below, but it is traditional, and there are many results associated with it.
Similarly, a space is Dieudonné-complete if its underlying topology is completely uniformizable?. According to Wikipedia, the term ‘topologically complete’ is sometimes used for this concept instead, which is just asking for trouble.
A compact space is necessarily complete. A space is called precompact if its completion (see below) is compact. For metric spaces (or even uniform spaces), there is a natural notion of a totally bounded space; in classical mathematics, we have the theorem that a space is totally bounded if and only if it is precompact. Similarly, a space is compact if and only if it is both complete and totally bounded (or more generally, both complete and precompact). Thus the purely topological property of compactness is the conjunction of the nontopological properties of completeness and total boundedness.
In some constructive approaches to analysis (particularly Bishop's school), this is taken as the definition of compactness, because it holds of examples such as the unit interval that fail to be compact (in the usual sense) without the fan theorem. However, in this case, compactness is no longer a topological property?. This is reconciled somewhat with the theory of localic completion, in which a uniform space is totally bounded if and only if its localic completion is compact (in the usual sense).
Every complete metric space is a Baire space. Since being a Baire space is a topological property, it follows that every topologically complete space is a Baire space.
Are there (necessarily nonmetrizable) complete spaces that are not Baire spaces?
The set of Cauchy filters on a space may naturally be given the same sort of structure as itself has, and this space will be complete. Exactly how to do this depends on what structure is supposed to have, of course, and one can make the general statement false by requiring something artificial as the structure in question, most extremely the structure of being a specific non-complete space. But it works for most natural categories of spaces.
The general idea is this: every point in generates a principal ultrafilter (consisting of those sets to which the point belongs), so there is a natural map from to . Furthermore, this map is a morphism of the appropriate structure, which in particular makes it Cauchy-continuous (preserving Cauchy filters) and continuous (preserving limits). So all of the limits in still exist in , but now each Cauchy filter in (having become both a Cauchy filter in and a point in ) has a limit as well. The additional Cauchy filters based on the additional points in will also have a limit in , essentially because is a monad (so a Cauchy filter of Cauchy filters folds into a single Cauchy filter).
There is a problem that is rather larger than necessary; for example, all of the filters that converge to a given point in (not just the free ultrafilter at that point) exist in and converge to one another. But you can take a quotient of to make it Hausdorff, obtaining the Hausdorff completion of . In case was not Hausdorff to begin with, one can sometimes also force the quotient to leave in just as much redundancy as has but no more, obtain a straight completion of . But really, it's most natural to make the space Hausdorff at the same time.
Details to come, if I get around to it.
We have a picture like this, where is the original space, gives a Hausdorff quotient, and an overline indicates completion:
(Here the arrows are drawn horizontally to put styles on them; they should all be diagonal in the only possible way.)
At least if is a metric space, then we can also construct its completion as a locale, the localic completion, whose spatial part is the above space, but which in constructive mathematics may not be spatial. This is useful to have even if is already complete.
When Bill Lawvere interpreted (in Lawvere 1973) metric spaces as certain enriched categories, he found that a metric space was complete if and only if every adjunction of bimodules over the enriched category is induced by an enriched functor. Accordingly, this becomes the notion of Cauchy complete category. (Note that one must say ‘Cauchy’ here, since this is weaker than being a complete category, which is based on an incompatible analogy.)
A list of classical concepts of completeness in topology (including some not listed here) is at
A.V. Arkhangel′skii (1977). Complete space. Matematicheskaya entsiklopediya. Updated English version.