This question arose more from curiosity than from an actual problem. There are situations when you embed some space $X$ in a set of filters on $X$, which inherits properties of $X$ or has even better properties. The prominent examples are (and in fact the only ones I am aware of):

(1) $X$ is a topological space and we consider the Stone-Cech compactification $\beta X$, which can be constructed from the set of all ultrafilters on $X$. We gain the obvious advantage of compactness.

(2) $X$ is a uniform space, and the completion of $X$ can be constructed from a set of Cauchy filters on $X$.

In some examples algebraic structures can be preserved. For example when $X$ is a a SIN topological group, the completion is a complete topological group.

The question is:

What other examples with an interesting application of spaces of filters like in the above examples are there?

For a motivation consider the Stone-Cech compactification $\beta \mathbb{N}$ of the natural numbers, which has a natural structure of a (noncommutative) monoid with a multiplication, which is only left continuous. Nevertheless it can be used to sho interesting results, for example about IP-sets. (http://en.wikipedia.org/wiki/IP_set)

(I hope this question is not too vague, in order to qualify as a real question.)