Suppose CH holds and $\mathbb{P}$ is a poset of size $\omega_1$, such that forcing with $\mathbb{P}$ preserves $\omega_1$.  Does forcing with $\mathbb{P}$ preserve CH?  If $\mathbb{P}$ is proper then the answer is yes, see [this question.][1]  Is this true for improper $\omega_1$-preserving posets of size $\omega_1$?


  [1]: https://mathoverflow.net/questions/53452/iterated-forcing-and-ch