Does a neighborhood of a point include that point? I'm working through the topology problem set in baby Rudin and I'm wondering if a neighborhood $N_r(p)$ of a point $p$ in a metric space automatically contains $p$, or just $\forall q|d(p,q)<r$, but not necessarily $p$.
My intuition says that because $d(p,p)=0$ and $0<r$, $p$ should automatically be in any neighborhood of $p$, but I wanted to verify that I was correct.

Yes it does. From That definition we are are talking about all points within a radius $r$ of $p$.
