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.](https://mathoverflow.net/questions/53452/iterated-forcing-and-ch) Is this true for improper $\omega\_1$-preserving posets of size $\omega\_1$?



Does small forcing preserve CH?
Yes. Let $X$ be a name for a subset of $\omega$. It can be described the following way: for every $i<\omega$ $\{p^i\_\alpha:\alpha<\omega\_1\}$ a maximal antichain, $\{\varepsilon^i\_\alpha:\alpha<\omega\_1\}$ where $\varepsilon^i\_\alpha\in\{0,1\}$ and $p^i\_\alpha$ forces $i\in X$ iff $\varepsilon^i\_\alpha=1$. In $V[G]$ the function $i\mapsto h(i)$ is an $\omega\to\omega\_1$ map where $p^i\_{h(i)}\in G$. As $P$ preserves $\omega\_1$, the range of $h$ is bounded, i.e., some $p$ forces that $h(i)<\gamma$ for all $i$, for some $\gamma<\omega\_1$. Now the structure $\langle p,\langle p^i\_\alpha,\varepsilon^i\_\alpha:\alpha<\gamma\rangle\rangle$ fully describes $X$. 


