# Changes

* A [[topological manifold]] $M$ (in particular, $M$ is Hausdorff and second-countable)
* An [[atlas ]] of coordinate charts $\varphi_i:U_i \to V_i, i \in I$ from $M$ to $\R^n$ (in other words an open cover $U_i$ of $M$ with homeomorphisms from each member $U_i$ of the open cover to open sets $V_i$ in $\R^n$)
satisfying the compatibility condition: the transition function between any two coordinate charts of the atlas is a diffeomomorphism of open subsets of $\R^n$. In symbols: