Closed Lindel\"of $\Sigma$ Kernels of Neighborhood Assignments in Charming Spaces

arXiv:2610.35909v1 Announce Type: new Abstract: Every neighborhood assignment on a charming space $X$ has a countable kernel whose closure in $X$ is Lindel\"of $\Sigma$. Consequently, for any two neighborhood assignments $M$ and $N$ on $X$, there is a closed discrete set $D\subseteq X$ 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 $\mathrm{HFC}_{\mathrm w}$ subspace of $[\omega_1]^{\leq\omega}$ is dually closed countable but not charming. For a fixed Lindel\"of $\Sigma$ 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 $\kappa$ and every $\gamma<\kappa^+$, there is a compact example with a dense $\sigma$-compact Lindel\"of $\Sigma$ kernel $Y$ satisfying $hL(Y)=|Y|=\kappa$ whose first stabilization stage is $\gamma$.

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