Inversion is uniformly stable on compact chart families

Identify $M_4(\mathbb{R})$ with ${\mathbb{R}}^{16}$ and write $d_{\mathrm{\infty }}(A,B)={\max }_{a,b}\mathrm{\mid }{A}_{ab}-B_{ab}\mathrm{\mid }$. The inequalities $d_{\mathrm{\infty }}(A,B)\le \mathrm{\parallel }A-B{\mathrm{\parallel }}_2\le 4d_{\mathrm{\infty }}(A,B)$ show that maximum-entry and Euclidean neighborhoods give the same topology. Each entry projection is continuous since its change is at most $d_{\mathrm{\infty }}(A,B)$. Thus [F4] and [F5] make determinant continuous. At a matrix $A$ with nonzero determinant, continuity gives a neighborhood where $\mathrm{\mid }\det B-\det A\mathrm{\mid }<\mathrm{\mid }\det A\mathrm{\mid }\mathrm{/}2$, so the determinant stays nonzero. By [F4], ${\mathrm{G}\mathrm{L}}_4(\mathbb{R})$ is therefore open, independently of inversion.

Apply [F1] to the open set ${\mathrm{G}\mathrm{L}}_4(\mathbb{R})$ from step 1.1. Inversion is continuous there, also for $d_{\mathrm{\infty }}$. Consequently, for every invertible center matrix $A$ and the given $\eta >0$, some $r>0$ has the following property: $d_{\mathrm{\infty }}(B,A)<2r$ implies that $B$ is invertible and $d_{\mathrm{\infty }}(B^{-1},A^{-1})<\eta \mathrm{/}2$. Indeed, intersect an open neighborhood of $A$ contained in ${\mathrm{G}\mathrm{L}}_4(\mathbb{R})$ with the inverse image under inversion of the radius-$\eta \mathrm{/}2$ ball about $A^{-1}$, and take a sufficiently small ball in that intersection.

For each coordinate domain $U_j$, use all pairs $(x,r)$ with $x\in K_j$ and $r>0$ satisfying step 2.1 for $A=G_j(x)$. The sets $V_{j,x,r}=\{y\in U_j:d_{\mathrm{\infty }}(G_j(y),G_j(x))<r\}$ are open in $U_j$ by continuity of $G_j$ and cover $K_j$. This uses neighborhoods in the coordinate domain, without identifying that domain itself with a Euclidean open set. Using all admissible pairs requires no selection of a radius at every point.

If every $K_j$ is empty, take $\delta =1$. Otherwise, [F2] gives a finite indexed subcover of each nonempty $K_j$ from step 3.1. Set $\delta >0$ to the minimum of the associated finitely many positive radii over the finitely many $j$. For $y\in K_j$ choose a selected $V_{j,x,r}$ containing it. Then $d_{\mathrm{\infty }}(G_j(y),G_j(x))<r$ and $d_{\mathrm{\infty }}({\tilde{G}}_j(y),G_j(x))<\delta +r\le 2r$. Step 2.1 makes both matrices invertible and puts both inverses within $\eta \mathrm{/}2$ of $G_j(x{)}^{-1}$. The triangle inequality gives $d_{\mathrm{\infty }}({\tilde{G}}_j(y{)}^{-1},G_j(y{)}^{-1})<\eta$. Empty sets impose no restriction, and no continuity or derivatives of the perturbation were used.