The regular domain of the Kerr metric is open

For all real $M,a$ and every point $x\in\mathbb R^4$: if $x$ lies in the regular domain

then every point of some neighbourhood of $x$ (product topology on $\mathbb R^4$) lies in the regular domain as well.

This openness statement is what allows Layer II to pass from the generic Christoffel symbol equals the closed form at every regular point to differentiability of the generic Christoffel symbol at $x$: the two functions agree on a whole neighbourhood, so their slice derivatives coincide.

Formalization Note Stated as ∀ᶠ y in 𝓝 x, RegKerr M a y.