Ergodic theory question about the support of a measure. I am trying to work through some problems from  " Ergodic theory with view towards number theory" but I am finding it a tad more difficult than expected. In particular, I have the following question: 

For $X$ a compact metric space, $T:X\to X$ continuous and $\mu$ a $T$-invariant ergodic probability measure defined on the Borel sets of $X$. Prove that for $\mu$ almost every $x\in X$ and $y\in \operatorname{Supp}(\mu)$ there exists a sequence $n_k\nearrow\infty$ such that $T^{n_k}(x)\to y$. Recall
  $$\operatorname{Supp}(\mu)=X-\bigcup U$$
  where $U\subset X$ is open and has null measure.

This problem screams the Poincaré Recurrence Theorem to me, but I am not sure where to use it. I could really use some guidance for this problem.
NOTE: This is not homework for a class, I just really want to learn ergodic theory.
 A: *

*It is enough to prove that for each $y\in\operatorname{Supp}\mu$, we have 
$$\mu\{x\in X\mid\exists n_k\uparrow +\infty, T^{n_k}x\to y\}=1.$$
Indeed, assume we have done that. The support of $\mu$ is a closed set in a separable metric space, hence it is separable. Let $\{y_j,j\in\Bbb N\}$ be a countable dense subset. As a countable union of negligible sets is negligible, we have 
$$\mu\left\{x\in X\mid\forall j\in\Bbb N,\exists n_k\uparrow +\infty, T^{n_k}x\to y_j\right\}=1.$$
Let $y\in \operatorname{Supp}(\mu)$. Then for each $i$, consider $y_{j_i}$ for which $d(y,y_{j_i})<i^{-1}$, and construct $(n_i)$ by induction, taking $d(y_{j_i},T^{n_i}x)<i^{-1}$. 

*Fix now $y\in\operatorname{Supp}\mu$, and call $C:=\left\{x\in X\mid\exists n_k\uparrow +\infty, T^{n_k}x\to y\right\}$. We can check that $C$ is almost invariant, in sense that $\mu(C\Delta T^{-1}C)=0$. As $T$ is ergodic, the measure of $C$ is either $0$ or $1$. 

*As $y$ is in the support of $\mu$, for each $l$, $\mu\left(B\left(y,l^{-1}\right)\right)$ is positive. By Poincaré recurrence theorem,  $\mu\left( \left\{x\in B\left(y,l^{-1}\right) \mid T^nx\in  B\left(y,l^{-1}\right)\mbox{ for infinitely many }n\mbox{'s}   \right\}  \right)$ is positive hence, by ergodicity, its value is $1$. Therefore, the measure of $C$ is $1$.
A: Realize that it is enough to prove the following proposition:
(P1) For $\mu$ almost every $x \in X$, when you pick any neighborhood $U$ around any point in the support, we have that $x$ visits $U$ infinitely many times (i.e., $T^{i} x \in U$ for infinitely many $i \ge 0$).
This kind of realization where the strictly increasing sequence $(n_k)_k$ is dealt away from the beginning is a standard trick when dealing with things like omega limits..
Let's prove P1. Suppose $y$ is some arbitrary point in the support and $U$ is some neighborhood of $y$. Consider the following proposition about $U$:
(P($U$)) For $\mu$ almost every $x \in X$, $x$ visits $U$ infinitely many times.
The proposition P($U$) is true because $U$ has positive measure and $\mu$ is ergodic. (Use the ergodic theorem. If you want to avoid using the ergodic theorem at this point, use the fact that $x$ must visit $U$ at least once and the fact that visiting $U$ once is enough to guarantee infinitely many visits (this is Poincare recurrence theorem))
Since $y$ and $U$ were arbitrary, can we conclude that P1 is true? We just need to tackle the problem that there could be more than countably many choices for $y$ and $U$. But notice that the proposition P($U$) makes no reference to $y$, so uncountably many possibilities for $y$ is not a problem. We only need to deal with uncountably many possibilities for $U$.
Since $X$ is second countable, we have a countable collection $\mathcal U_{\mu}$ such that each element in the collection is a neighborhood of some point in the support and that for each $y$ in the support and each neighborhood $U$ of $y$, there is $U' \in \mathcal U_{\mu}$ such that $y \in U' \subset U$.
To prove P1, it is enough to prove P2:
(P2) For $\mu$ almost every $x \in X$, for every $U \in \mathcal U_{\mu}$, $x$ visits $U$ infinitely many times.
But since $\mathcal U_{\mu}$ is countable, P2 is equivalent to P3:
(P3) For every $U \in \mathcal U_{\mu}$, for $\mu$ almost every $x \in X$, $x$ visits $U$ infinitely many times.
But P3 is something that we already proved.
