= Club agreement of countable filtrations
{title2=$f[F_\delta]=G_\delta\quad\text{on a club}$}
For two <kappa-filtrations> at $\kappa=\omega_1$ exhausting sets of size $\aleph_1$, any bijection between the sets carries corresponding stages onto one another on a <club set> of indices. Bound both forward and inverse images at each countable stage by a function $g:\omega_1\to\omega_1$. The limit ordinals closed under all earlier $g$-values form a club; continuity gives equality at those ordinals.
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