I was rereading an answer to an old question of mine and it included a reference to the fact that $2^{\omega_1}$ was separable. I'm having a hard time finding a reference for this fact, and the proof is not immediately obvious to me. Can anyone provide me with a cite and/or a proof?

Should have searched a bit harder before asking this one. This is an immediate consequence of the HewittMarczewskiPondiczery theorem: Let $m \geq \aleph_0$. If $\{X_s : s \in S\}$ are topological spaces with $d(X_s) \leq m$ and $S \leq 2^m$ then $d(\prod_s X_s) \leq m$. 


This is indeed the HewittMarczewskiPondiczery theorem. My proof, following Engelking, is here. It's in fact not that hard, the fact for a product of copies of 2 point discrete spaces already implies the general theorem pretty quickly. 

