Definition (Chart)

Let $S$ be a set. A chart for $S$ is a pair $(\mathcal{U},\phi)$ with

1. $\mathcal{U}$ a subset of $S$, and
2. $\phi:\mathcal{U}\to \mathbb{R}^{n}$ and injection for which $\phi(\mathcal{U})$ is an open subset of $\mathbb{R}^{n}$.