Smooth atlas

Set

 context $\langle M,T\rangle$ … second-countable Hausdorff space context $n\in \mathbb N$ definiendum $A\in$ it inclusion $A\subseteq$ atlas ($\langle M,T\rangle,n$) forall $\langle V,\phi\rangle,\langle W,\psi\rangle\in A$ postulate $\phi\circ\psi^{-1}$ … smooth

A priori “$\phi\circ\psi^{-1}$” in the postulate doesn't make sense as their domains/codomains will not much. Here, really, one must choose functions with appropriately restricted domain.

Universal property

For a smooth atlas there are smooth coordinate changes on $\mathbb R^n$.

Idea

For a smooth atlas there are smooth coordinate changes on $\mathbb R^n$.