Characterization of Lindelöf finite and countable Hattori powers

Characterize, for each fixed integer n≥4, the subsets A of the real line for which the Hattori power H(A)^n is Lindelöf, and characterize the subsets A for which the countable power H(A)^ω is Lindelöf; in particular, determine structural conditions on the Sorgenfrey subspace B=ℝ\A that explain Kulesza’s announced even-to-odd Lindelöf implication and possible odd-to-even failures.

Background

The paper proves in ZFC that Lindelöfness of the square H(A)^2 implies Lindelöfness of the cube H(A)^3, and it characterizes this condition by the absence of a Cantor subspace in the Sorgenfrey stratum B=ℝ\A. It also establishes hereditary-Lindelöf characterizations for finite and countable powers in terms of Lindelöfness of finite Sorgenfrey powers of B.

The results do not characterize ordinary Lindelöfness for powers H(A)^n with n≥4 or for H(A)^ω. A ZFC construction shows that H(A)^2 and H(A)^3 can be Lindelöf while H(A)^4 and all higher finite powers are not Lindelöf, so the square-to-cube criterion does not extend directly to higher powers. The question asks for structural conditions on B that would yield complete characterizations and clarify the general even-to-odd implication announced by Kulesza, together with possible odd-to-even separations.