A topological space $X$ is called a **$\sigma$-space** if every $F\_{\sigma}$-subset of $X$ is $G\_{\delta}$.


A topological space $X$ is called a **$Q$-space** if any subset of $X$ is $F\_{\sigma}$.


**Definition.** A topological space $X$ is called a **hereditary $\sigma$-space** if every subset of $X$ is $\sigma$-space.


**Question.** Is there a hereditary $\sigma$-space $X$ such that it is not $Q$-space ?



Is there a hereditary $\sigma$-space $X$ such that it is not $Q$-space?
Every $S\_1(B\_\Gamma,B\_\Gamma)$ space is a $\sigma$-space, and the property $S\_1(B\_\Gamma,B\_\Gamma)$ is hereditary for subsets (B. Tsaban and M. Scheepers, [The combinatorics of Borel covers](https://doi.org/10.1016/S0166-8641(01)00078-5), Topology and its Applications 121 (2002), 357-382.)


For example, a Sierpiński set satisfies $S\_1(B\_\Gamma,B\_\Gamma)$.


On the other hand, it is very difficult to be a Q-set. For example, Q-sets have Lebesgue measure zero. In particular, they cannot be Sierpiński.


