Čech-Stone compactification of $\mathbb N$ and ultrafilters on $\mathbb N$ I have found in the literature that the Čech-Stone compactification $\beta\mathbb N$ of $\mathbb N$ (or more generally, of any discrete topological space) can be identified with ultrafilters on $\mathbb N$. Under this identification, the natural numbers correspond to the principal ultrafilters.
How exactly does the identification of $\beta\mathbb N$ and the ultrafilters on $\mathbb N$ look like? How does the topology on the set of ultrafiltres on $\mathbb N$ look like?

The object $\beta\mathbb N$ is notoriously elusive. It is an important object of study in set theoretic topology and is highly sensitive to various set-theoretic axioms. Generally speaking, non-principal ultrafilters are very elusive, their general existence requiring a weak form of the axiom of choice. Thus, it is very difficult to imagine what ultrafilters look like, let alone what $\beta \mathbb N$ looks like. 
One thing for instance is that one can extend the addition of natural numbers to an addition operation on $\beta \mathbb N$ but it is not commutative. You can further read quite a lot on $\beta \mathbb N$ on any text on set-theoretic topology, but as said, this is far from conclusive or straightforward. 
