# 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 topology on the set $\mathcal U$ of ultrafilters has basic open sets of the form $B_X=\{u\in\mathcal U:X\in u\}$ where $X\subseteq\mathbb N$. – bof Jan 30 '14 at 10:23

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.
• Do you mean then to ask how to prove that $\beta \mathbb N$, as the set of ultrafilters on $\mathbb N$, is the required compactification? if so, then any text introducing it will answer that. – Ittay Weiss Jan 30 '14 at 9:43
• I would like to see the prescription of the mapping which identifies $\beta\mathbb N$ with the ultrafilters. – user124187 Jan 30 '14 at 9:51