In Topology, What is a Coordinate System?
Definition
Let us consider $M$ to be a $n$-dimensional manifold. Suppose two open sets $U\subset M$, $\tilde{U} \subset \mathbb{R}^n$ and a homeomorphism $\phi\ :\ U \rightarrow \tilde{U}$ are given. Then, the ordered pair $(U, \phi)$ is called the coordinate system on $M$, or simply coordinates$(\mathrm{Chart})$.
Explanation
If $p \in U$, $\phi (p)=0$, then $(U,\phi)$ is called the center in $p$. Moreover, $U$ is referred to as the coordinate domain or the coordinate neighborhood. If $\phi (U)$ is an open ball in $\mathbb{R}^n$, then $U$ is called the coordinate ball. $\phi$ is known as the coordinate map, and when emphasizing that $U$ is not the entire set, it is called the local coordinate map. Similarly, for the chart $(M,\phi)$, $\phi$ is referred to as the global coordinate map.