2026/07/28 by Xing-Yu Hu
#math.LO #math.GN
For a ladder-system space XL with carrier S⊆ Eω1ω, the finite-label uniformization property M<ω characterizes countable metacompactness, and countable metacompactness is equivalent to the Δ-property. Both equivalences are known for stationary carriers. For arbitrary carriers, an active-tail formulation gives a direct proof that M<ω is equivalent to the Δ-property and leads to a support-finite decomposition theorem, together with club-smallness and trace criteria that avoid explicit ladder-position thresholds. A club-gap argument, combined with Fodor's lemma, shows that finite and countable tail multiplicity determine the same carrier ideal, namely NS\restriction S. Subsets of the isolated part that meet each ladder in only finitely many points have clopen remainder traces, and these traces form a generalized Boolean algebra. All such traces are disjoint from the carrier part of the remainder. Finally, the subcarriers whose restricted spaces are σ-closed discrete form an ideal CL containing NS\restriction S. If XL is a Δ-space, a threshold-based gluing argument shows that CL is a σ-ideal. Whether this holds for every ladder system remains open.