Is the Katětov extension of $\Bbb N$ zero-dimensional? I take $X=\kappa\Bbb N$ to be $\Bbb N\cup\{p:p \text{ is a free ultrafilter on }\Bbb N\}$.  Each singleton in $\Bbb N$ is open and a local base at any free ultrafilter $p$ is given by $\{\{p\}\cup A:A\in p\}$.

Is X zero-dimensional?

In other words, is it true that each point of $X$ has a neighborhood basis of clopen sets?
It is claimed for example here that each basic open set $\{p\}\cup A$ is clopen.  

I am not sure about this claim, as it seems that from basic properties of ultrafilters one can prove it is not the case.
Fact: Given an infinite subset $A$ of $\Bbb N$ there is a free ultrafilter containing $A$.
(Take the set of all cofinite subsets of $A$.  That forms a filter basis that generates a free filter on $\Bbb N$.  Any ultrafilter that extends that filter will be a free ultrafilter containing $A$.)
Now take a free ultrafilter $p\in X$ and an (open) neighborhood $U=\{p\}\cup A$ with $A\in p$.  I claim that $U$ is never closed in $X$.  $A$ must be infinite because $p$ is free.  Partition $A$ into two infinite sets: $A=B\cup C$. By standard ultrafilter properties exactly one of the two subsets, say $B$, must be in $p$.  By the Fact above, there is an ultrafilter $q$ containing $C$, and $q$ is necessarily distinct from $p$.  This $q$ is in the closure of $U$.  Indeed, take any neighborhood $\{q\}\cup D$ with $D\in q$.  $D\cap C\in q$ so $D\cap C$ is not empty and $D$ meets $A$, so any neighborhood of $q$ meets $U$.  This shows that $U$ is not clopen.
Can you see anything wrong with this argument?

Added: My original question was the one above about $\kappa\Bbb N$, but I also mistakenly thought $\kappa\Bbb N$ was the same as the Cech-Stone compactification $\beta\Bbb N$.  Thanks @EricWofsey for setting me straight.
 A: Your argument is correct, but the space you describe is not the Stone-Čech compactification!  Indeed, you can see quite quickly that it is not compact: for each free ultrafilter $p$, the set $\{p\}\cup\mathbb{N}$ is open, and these form an open cover with no finite subcover.
For the Stone-Čech compactification, the basic open sets are those of the form $U_A=A\cup\{p:A\in p\}$ for $A\subseteq\mathbb{N}$.  (Or, identifying points of $\mathbb{N}$ with the principal ultrafilters, you just take the set of all ultrafilters that contain $A$.)  It is immediate that these are clopen, since the complement of $U_A$ is just $U_{\mathbb{N}\setminus A}$.  (Of course, it is not immediate that this space really is the Stone-Čech compactification of $\mathbb{N}$, but that's a longer story.)
A: $\kappa \Bbb N$ cannot be zero-dimensional, it then would be a regular $H$-closed space and thus (by standard results) compact, which your space is not (as $\kappa \Bbb N\setminus \Bbb N$ is infinite, closed and discrete).
