Charts
AN OPEN SET AND THE MAP THAT FLATTENS IT
A chart is a pair: an open set \(U\) of a topological space, and a map \(x\) that is a homeomorphism from \(U\) onto an open subset of \(\mathbb{R}^d\). Choosing a chart is what turns a point into numbers: \(x(p)\) is a \(d\)-tuple. Another chart over the same point may return a different tuple.
The definition
Let \((M,\mathcal{O})\) be a \(d\)-dimensional topological manifold. A chart of \((M,\mathcal{O})\) is a pair \((U,x)\) with \(U \in \mathcal{O}\) and \[ x : U \to x(U) \subseteq \mathbb{R}^d \] a homeomorphism onto an open subset of \(\mathbb{R}^d\). The map \(x\) is the chart map, and its components \[ x^1, \dots, x^d : U \to \mathbb{R}, \qquad x(p) = \big(x^1(p), \dots, x^d(p)\big) \] are the coordinate maps of the chart. The numbers \(x^1(p), \dots, x^d(p)\) are the coordinates of \(p\) with respect to \((U,x)\).
The chart map is one map; the coordinate maps are its components
\(x\) lands in \(\mathbb{R}^d\), so its value at a point is a \(d\)-tuple of real numbers, and \(x^i\) is the map returning the \(i\)-th of them. There are \(d\) coordinate maps and one chart map, and they carry the same information: \(x\) is determined by all the \(x^i\) together. The superscript is an index, not an exponent.
Each \(x^i\) is a real-valued map on \(U\), which is what makes coordinates usable as numbers: a function of position on the space becomes a function of \(d\) real variables by composing with \(x^{-1}\).
Coordinates belong to the chart, not to the point
The point \(p\) is an element of a set. It has no numbers attached to it, and the space it lives in has no preferred way to produce any. A chart is a choice, and \(x(p)\) is what that choice returns.
Where two charts \((U,x)\) and \((V,y)\) both contain \(p\), both \(x(p)\) and \(y(p)\) are defined, and there is no reason for them to be equal: \(x\) and \(y\) are separately chosen homeomorphisms, onto open subsets of \(\mathbb{R}^d\) that need have nothing to do with one another. The point has not moved and the space has not changed. What changed is which map was applied to it.
The domain of a chart is also a choice. \(U\) is required to be open and to be carried homeomorphically onto an open subset of \(\mathbb{R}^d\); it is not required to be large, and no particular point is required to lie in it. Outside \(U\), the chart says nothing at all.
One chart, run by hand
Let \(M\) be the unit circle: the points \((\cos t, \sin t)\) in \(\mathbb{R}^2\), with the subset topology. Take \(U\) to be the circle without the point \((1,0)\), and let \(x\) send each point of \(U\) to its angle, measured anticlockwise from \((1,0)\). Then \(x(U) = (0, 2\pi)\), an open interval, and \(x\) is invertible and continuous in both directions, so \((U, x)\) is a chart with \(d = 1\).
At the point \(q = (0,-1)\), the bottom of the circle, \(x(q) = 3\pi/2\). A second chart \((V, y)\), with \(V\) the circle without \((-1,0)\) and \(y\) the angle from \((1,0)\) taken in \((-\pi, \pi)\), returns \(y(q) = -\pi/2\). One point, two numbers, and nothing makes either of them the right one.