Just to fix the environment, let's work in the Baire space $\omega^\omega$, the space of infinite sequences of natural numbers with the product of the discrete topology over $\omega$. We say that a subset $A\subseteq \omega^\omega$ satisfies Hurewicz dichotomy if either it's $F\_\sigma$ or there exists a Cantor set (a closed subset with no isolated points) $\mathcal{C}\subseteq \omega^\omega$ such that $\mathcal{C}\setminus A$ is countable dense in $\mathcal{C}$.  

Now Hurewicz proved that



> 
> Every analytic subset $A\subseteq\omega^\omega$ satisfies the Hurewicz dichotomy.
> 
> 
> 


Now my questions are:


1. What is the consistency strength of the statement "Every subset of the Baire space satisfies the Hurewicz dichotomy"?
2. What is the relationship between the Hurewicz dichotomy property and the perfect set property (also consistency-wise)?


Thanks!



Consistency of the Hurewicz dichotomy property
The anwer is in: *Tall, Franklin D.; Todorcevic, Stevo; Tokgöz, Seçil*, [**The strength of Menger’s conjecture**](http://dx.doi.org/10.1016/j.topol.2020.107536).


In the paper they prove (among other things) that the Hurewicz dichotomy extended to all subsets of the reals is equiconsistent with an inaccessible cardinal.


