For a "reasonable" pointclass ${\bf \Gamma}$, say that a second-countable space $(X,\tau)$ is ${\bf \Gamma}$-describable iff for some (equivalently, every) enumerated subbase $B=(B_i)_{i\in\omega}$ we have $$\{\langle f,g\rangle\in (2^\omega)^2: \bigcup_{f(i)=1}B_i=\bigcup_{g(j)=1}B_j\}\in{\bf\Gamma}.$$
For example, Cantor space is properly ${\bf \Pi^0_2}$-describable and Baire space is properly ${\bf \Pi^1_1}$-describable. More interestingly, at the end of this earlier question of mine I gave an easy construction of spaces which are properly ${\bf \Pi^0_\alpha}$-describable for arbitrarily large countable $\alpha$. However, these spaces were pretty terrible and I'm interested in finding "tamer" examples:
Is there a second-countable Hausdorff space which is properly ${\bf \Pi^0_\alpha}$-describable for some countable $\alpha>2$?
The examples mentioned above were not even $T_1$.
