Kappa-filtration ($(A_{\alpha }{)}_{\alpha <\kappa }$)

For a set $A$ of size an uncountable regular cardinal $\kappa$, an increasing sequence $(A_{\alpha }{)}_{\alpha <\kappa }$ with union $A$, each $|A_{\alpha }|<\kappa$, and $A_{\delta }={\bigcup }_{\alpha <\delta }{A}_{\alpha }$ at every nonzero limit stage.

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.