You hear the term coordinate system thrown around a lot, and we all know the usual examples (polar coordinates in $\mathbb{R}^2$, spherical coordinates in $\mathbb{R}^3$, etc.), but in truth I have no idea what the term actually means.
Is there a rigorous definition of "coordinate system"?
In particular, if I were to write "Let $c : C \rightarrow X$ denote a coordinate system for $X$," what kind of objects are $C$ and $X$ (affine spaces? topological spaces? something else?), and what kind of entity is $c$ (a surjective function? a continuous mapping? something else?)
A "coordinate system" or more precisely a "local coordinate system" in this context is also known as a chart. A chart for a topological space $X$ is a continuous map $\phi:U\to X$ which is a homeomorphism onto its image, where $U$ is an open subset of $\mathbb{R}^n$. The coordinate functions $x_1,\ldots,x_n$ on $U\subset\mathbb R^n$ then give rise to coordinate functions $x_i\circ\phi^{-1}$ on $\phi(U)\subset X$.