Normal neighborhood and normal coordinate chart

Because ${\exp }_p{\mathrm{\mid }}_D$ is a diffeomorphism, its inverse is a well-defined smooth map $U\to D$. The supplied basis makes $E_e$ a specified linear isomorphism, so $E_e^{-1}(D)$ is open and $x_e$ is a diffeomorphism onto it. Thus the intrinsic normal neighbourhood does not depend on coordinates, while the displayed coordinate list records exactly its dependence on the supplied basis.

Star-shaped means that $v\in D$ implies $tv\in D$ for every $0\le t\le 1$, including both endpoints. The zero vector belongs to $D$ and maps to $p$. In dimension zero, the ordered basis is the empty tuple, ${\mathbb{R}}^0$, $T_{p}M$, $D$, and $U$ are singletons, and the formula gives the unique chart; dimension one is literal. If $M$ is empty there is no centre $p$. The basis is supplied rather than chosen, so the only choice principle is the stated ${\mathrm{A}\mathrm{C}}_{\omega }$ inherited through [F1].