It is easy to see a forcing of size $\aleph\_1$ is proper if and only if is semiproper. I was wondering when such an equivalency holds between semi-proper and stationary-preserving forcings in $\rm ZFC$? Or consistently in a model where significant fragments of $\rm MM$ fail.



Properness for small forcings
The answer is no, as the following upcoming work of Shelah and Usuba shows:


**Theorem (Shelah-Usuba)**:
The following theories are equiconsistent with ZFC:


ZFC+CH+ “there is an $\omega \_1$-stationary preserving $\sigma$-Baire poset of size $\aleph\_1$
​ which is not semiproper”.


ZFC+“Martin’s axiom for semiproper posets of size $\aleph\_1$” + “there is an $\omega \_1$-stationary preserving $\sigma$-Baire poset of size $\aleph\_1$ which is not semiproper”.


ZFC+CH+“every $\omega \_1$-stationary preserving $\sigma$-Baire poset of size $\aleph\_1$ is semiproper”.


See [$\omega\_1$-Stationary preserving $\sigma$-Baire posets of size $\aleph\_1$](https://shelah.logic.at/papers/1010/).


