Abstract
Let H(A) be the Hattori space associated with A ⊆ R, and put B = R \ A. Kulesza announced at the 2018 Joint Mathematics Meetings that Lindelöfness of H(A)2n forces Lindelöfness of H(A)2n+1 for every positive integer n. The meeting abstract states this implication without a proof. This paper proves the case n = 1 in ZFC. H(A)2 is Lindelöf ⇒ H(A)3 is Lindelöf. The proof passes from an uncountable closed discrete subset of the cube to a perfect set with injective coordinate projections and uses the one-sided local bases to obtain three cocountable Pareto-maximality sets with countable triple intersection. The same one-sided geometry gives a pointwise half-space criterion for affine discreteness and, when H(A)2 is Lindelöf, a four-coordinate obstruction for uncountable closed discrete monotone traces. Known results on mixed Sorgenfrey–Euclidean products lead to hereditary-Lindelöf criteria for finite powers, a countable-power characterization, and a finite-power Lindelöf reduction. For every n ∈ N, the power H(A)n is hereditarily Lindelöf if and only if the Sorgenfrey power Bn is Lindelöf, and H(A)ω is hereditarily Lindelöf if and only if every finite Sorgenfrey power of B is Lindelöf. A transfinite construction gives A ⊆ R for which H(A)2 and H(A)3 are Lindelöf, whereas H(A)4 contains a closed discrete affine line of cardinality \( \mathfrak c. \) For this ZFC example, B2 is not Lindelöf and the second and third Hattori powers are not hereditarily Lindelöf.