Let $\mathbb{P}$ and $\mathbb{Q}$ be two forcing notions. Recall that we say $\mathbb{Q}$ is a subforcing of $\mathbb{P}$ if there exists a regular embedding $\mathbb{Q} \to \text{r.o.}(\mathbb{P}).$


**Question.** Let $\mathbb{P} \in \{Add(\omega, \kappa), Col(\omega\_1, \kappa) \}.$


(1) Is there a subforcing of $\mathbb{P}$ which is not Proper?


(2) Is there a subforcing of $\mathbb{P}$ which is not semi-Proper?


**Remark.** In [Subalgebras of Cohen algebras need not be Cohen](http://arxiv.org/abs/math/9610227), Koppelberg and Shelah show that there is a subalgebra of Cohen forcing which is not Cohen. 



Bad subforcings of nice forcing notions
No. A subforcing of a c.c.c. forcing is c.c.c. A subforcing of a countably closed forcing is countably-strategically-closed, which implies proper. (This is easy to see via countable elementary submodels. Use the strategy to construct a generic condition.)


Furthermore, every subforcing of a proper forcing is proper. Properness is equivalent to preserving stationary subsets of $[\kappa]^\omega$ for all $\kappa$. The stationarity of $X \subseteq [ \kappa ]^\omega$ cannot be restored once killed, since killing it is just adding some $f : [\kappa]^{<\omega} \to \kappa$ such that no $x \in X$ is closed under $f$. (See Abraham's chapter of the Handbook.)


