Closed Lindelöf Σ Kernels of Neighborhood Assignments in Charming Spaces
Abstract
Every neighborhood assignment on a charming space X has a countable kernel whose closure in X is Lindelöf Σ. Consequently, for any two neighborhood assignments M and N on X, there is a closed discrete set D⊆ X such that M[N[D]]=X. We give an exact one-step formulation of the remaining D-space problem and identify the universal pointwise square-refinement condition with the classical z-space condition. For continuous closed surjections, we prove a fixed-assignment lifting equivalence. We also show that every uncountable HFCw subspace of [ω1]≤ω is dually closed countable but not charming. For a fixed Lindelöf Σ kernel Y, we iterate the Hu--Yin boundary reduction by a transfinite closed-core process. At every stage, the current iterate is a D-space if and only if X is a D-space. The iteration stabilizes before hL(Y)+ at the largest closed subspace in which the kernel trace and its complement are both dense. The bound is sharp: for every infinite cardinal κ and every γ<κ+, there is a compact example with a dense σ-compact Lindelöf Σ kernel Y satisfying hL(Y)=|Y|=κ whose first stabilization stage is γ.
Related
- On the closed neighborhood ideal of the square of broom and double broom graphs
- The v-number of t-closed neighborhood ideals
- Projective dimension of closed neighborhood hypergraphs via extended double covers
- Mean Lindelöf Hypothesis and Prime Geodesic Theorem on Bianchi Manifolds
- Lindelöf-type properties of spaces of Baire-one functions
BibTeX
@misc{hu2026closed,
title = {{Closed Lindelöf $Σ$ Kernels of Neighborhood Assignments in Charming Spaces}},
author = {Xing-Yu Hu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35909},
eprint = {2610.35909},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.35909},
}
RIS
TY - UNPB
TI - Closed Lindelöf Σ Kernels of Neighborhood Assignments in Charming Spaces
AU - Xing-Yu Hu
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.35909
UR - https://arxiv.org/abs/2610.35909
AB - Every neighborhood assignment on a charming space X has a countable kernel whose closure in X is Lindelöf Σ. Consequently, for any two neighborhood assignments M and N on X, there is a closed discrete set D⊆ X such that M[N[D]]=X. We give an exact one-step formulation of the remaining D-space problem and identify the universal pointwise square-refinement condition with the classical z-space condition. For continuous closed surjections, we prove a fixed-assignment lifting equivalence. We also show that every uncountable HFC_w subspace of [ω_1]^≤ω is dually closed countable but not charming. For a fixed Lindelöf Σ kernel Y, we iterate the Hu--Yin boundary reduction by a transfinite closed-core process. At every stage, the current iterate is a D-space if and only if X is a D-space. The iteration stabilizes before hL(Y)^+ at the largest closed subspace in which the kernel trace and its complement are both dense. The bound is sharp: for every infinite cardinal κ and every γ<κ^+, there is a compact example with a dense σ-compact Lindelöf Σ kernel Y satisfying hL(Y)=|Y|=κ whose first stabilization stage is γ.
ER -