TITLE: Different definitions of P-Points (ultrafilters)
QUESTION [6 upvotes]: Recently I have been exposed to the concept of P-Points. I have three definitions: 
A non-principal ultrafilter $u$ is a P-Point if:

For every sequence $\left < A_n \right >_{n\in \omega}$ of elements of $u$ there exists some set $B \in u$ that is almost contained in every $A_n$. (Most common one.)
For every $G_\delta$ that contains $u$, $u$ is in the interior of the $G_\delta$. (A bit hipster.)
For every function $f:\mathbb{N}\to \mathbb{N}$ there is some $A\in u$ such that $f| A$ is finite-to-one or constant. (Paper specific.)

Does anyone know how to prove that these definitions are equivalent? Or maybe could point out where some of the proofs may be?

REPLY [5 votes]: To see that (1) is equivalent to (2), all you really need to know is the following translation.  A basic open set in $\beta\mathbb{N}$ has the form $N_A = \{v : A\in v\}$, where $A$ is some subset of $\mathbb{N}$; and when $A \subseteq B$, $N_A\subseteq N_B$.  When $A$ is almost-contained in $B$, then we can only say $$N_A\cap (\beta\mathbb{N}\setminus\mathbb{N}) \subseteq N_B\cap (\beta\mathbb{N}\setminus\mathbb{N})$$ But since we're dealing primarily with non-principal ultrafilters, the proof still works out.
I like the hipster version.  (But I only learned about it after it was already cool.)