# lo.logic – Is there a Hausdorff space whose “covering problem” has intermediate complexity?

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 $$B=(B_i)_{iinomega}$$ we have $${langle f,granglein (2^omega)^2: bigcup_{f(i)=1}B_i=bigcup_{g(j)=1}B_j}in{bfGamma}.$$

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 $$T_1$$.