Lindelöf Powers of Hattori Spaces and the Perfect Set Property

Let H(A) be the Hattori space associated with A ⊆ R, and put B = R \ A. Hernández-Hernández, Ramírez-Chávez and Rojas-Hernández asked ([9], Problem 7.8) for characterizations of when H(A)³, all finite powers H(A)ⁿ, or H(A)^ω are Lindelöf. Using the square theorem and the perfect set property dichotomy, we show that if B has the perfect set property, then the following conditions are equivalent: B is countable, H(A) is second countable, H(A)² is Lindelöf, H(A)³ is Lindelöf, H(A)ⁿ is Lindelöf for every integer n ≥ 2, and H(A)^ω is Lindelöf. This applies in particular when B is analytic or Borel. We also show that the perfect set property assumption cannot be removed for higher powers. In ZFC there is a set A ⊆ R such that B has cardinality c and contains no subspace homeomorphic to 2^ω in the Euclidean topology, while H(A)² is Lindelöf and H(A)ⁿ is not Lindelöf for every n ≥ 4. In particular, H(A)^ω is not Lindelöf. The corresponding implication for the cube remains open.