A smooth manifold of dimension $n$ is a triple $(M,\tau,\mathfrak{A})$, where $M$ is a set, $\tau$ is a topology on $M$ that is Hausdorff and second-countable and $\mathfrak{A}$ is a maximal smooth atlas,
i.e. a collection of homeomorphisms $\varphi_i\colon U_i\rightarrow\varphi_i(U_i)$, indexed by some set $I$, where $U_i\subseteq M$ is open, $\varphi_i(U_i)\subseteq\mathbb{R}^n$ is open, such that $\varphi_i\circ\varphi_j^{-1}\colon\varphi_j(U_i\cap U_j)\rightarrow\varphi_i(U_i\cap U_j)$ is smooth for all $i,j\in I$, such that $\bigcup_{i\in I}U_i=M$ and such that this collection is maxi…