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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?

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  • $\begingroup$ 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 $X\subseteq\mathbb N$. $\endgroup$ – bof Jan 30 '14 at 10:23
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It might look like an obvious reference, but I consider the wikipedia survey on Stone–Čech compactification one of the best mathematical articles in wikipedia. In particular, it gives four different constructions of the Stone–Čech compactification, where most textbooks give only one or two constructions.

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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.

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  • $\begingroup$ Thank you, but this does not answer my question... $\endgroup$ – user124187 Jan 30 '14 at 9:41
  • $\begingroup$ 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. $\endgroup$ – Ittay Weiss Jan 30 '14 at 9:43
  • $\begingroup$ I would like to see the prescription of the mapping which identifies $\beta\mathbb N$ with the ultrafilters. $\endgroup$ – user124187 Jan 30 '14 at 9:51
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    $\begingroup$ See math.toronto.edu/~drorbn/classes/9293/131/ultra.pdf $\endgroup$ – Martín-Blas Pérez Pinilla Jan 30 '14 at 10:26

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