Square-to-Cube Lindelöfness and Power Separations in Hattori Spaces
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. 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. 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 c.
Related
- A Power-Diagram Upper Bound for Twenty-Six Variable-Radius Disks in a Square
- Singular difference graphs of vector spaces of square matrices
- BMO-Type characterizations and endpoint estimates for commutators of intrinsic square functions on Orlicz-Hardy spaces
- Nef Hattori classes and products of projective lines
- Applications of sequential spaces to Smyth power spaces and sober spaces
BibTeX
@misc{hu2026squaretocube,
title = {{Square-to-Cube Lindelöfness and Power Separations in Hattori Spaces}},
author = {Xing-Yu Hu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00853},
eprint = {2610.00853},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00853},
}
RIS
TY - UNPB
TI - Square-to-Cube Lindelöfness and Power Separations in Hattori Spaces
AU - Xing-Yu Hu
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.00853
UR - https://arxiv.org/abs/2610.00853
AB - 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. 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. 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 c.
ER -