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The definition of a coordinate chart is a map from an open set in a manifold to an open set in $\mathbb R^n$.

What is the motivation for requiring that the subsets in $\mathbb R^n$ be open? Without this requirement, for example, we could cover a compact manifold with one chart.

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    The whole manifold is both closed and open, so one chart is just fine. – 2017-02-03
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    I guess you need open sets to ensure that you can do calculus--define derivatives, etc. If you drop the requirement then what does it mean to say the coordinate map $\phi $ is a homeomorphism or, on a smooth manifold, a diffeomorphism? – 2017-02-03
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    The whole point of charts is to realize explicitly the "locally Euclidean" nature of the manifold – 2017-02-03

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Remember that charts are homeomorphisms from some chunk of the manifold to $\mathbb R^n$, not just any continuous function or even any open map. Here are a few things that go wrong if you try to use sets that are not open.

1) One of the fundamental constructions in geometry is that of the tangent space (I'll assume we're dealing with manifolds that have at least some amount of smoothness to them since this is tagged 'geometry'). A natural way of defining the tangent space is to look at curves that travel through a particular point on the manifold, and look at their derivatives in coordinates. In order to parameterize curves and talk about their derivatives, we need to be dealing with open sets, a fact you should be familiar with from vector calculus.

2) We want the change of coordinates to be smooth. That is to say, if we compose a chart and it's inverse and think of this as a map from $\mathbb R^n \to \mathbb R^n$, we want this map to be smooth. If one of the sets we parameterized by was not open, we could run into some serious problems. Here is a simple example. Suppose we had some surface, and we had a chart parameterizing some open, simply-connected patch in the surface, and another chart which parameterized some similar patch, but on the intersection, there was a punctured point in the parameterization, i.e. we parameterized everything but a point. Then the first map takes $\mathbb R^n$ up to the manifold and gives us coordinates, but the next map gives us back $\mathbb R^n$ minus a point. It is a standard fact from algebraic topology that there cannot be a homeomorphism between these sets, which means that we would have a lot of trouble dealing with such maps.

3) Another way that the change of coordinates maps can go afoul if the sets aren't open is if the parameterized patches intersect in only one point. Certainly this will not yield a map $\mathbb R^n \to \mathbb R^n$. What would it even mean for such a map to be smooth on the intersection, when the intersection isn't even the right topological dimension?

I'm sure there are many more sophisticated examples of when openness is needed. Rather than come up with lots of machinery that one would want to develop that depend on this, let's suffice to say that when dealing manifolds, any 'local' property can be dealt with by looking in charts and trying to do vector calculus, all of which depends on working in open sets.

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If the range of a chart map could be any subset of $\mathbb R^n$, not necessarily open, then any subset of $\mathbb R^n$ would be a manifold, covered by a single chart. Calling things like the Cantor set on the line (or a figure 8 in the plane, or the set of points in $\mathbb R^n$ with rational coordinates) manifolds would diverge wildly from the intended meaning of "manifold".