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Let $M$ be a topological manifold. If $(U, \phi)$ and $(V, \psi)$ are two coordinate charts, we say that these two charts are smoothly compatible if either

1) $U \cap V = \emptyset$

or

2) the transition map $\psi \circ \phi^{-1} : \phi(U \cap V) \rightarrow \psi(U \cap V)$ is a diffeomorphism.

From this definition, I am not getting the idea behind the concept of smoothly compatible. What does compatibility mean, here? The first condition seems to me even more vague as I don't understand how the empty intersection ensures smooth compatibility.

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Smoothly compatible means the following: Say you use one chart, and you want to transition to a different chart (perhaps a different parametrization turns out to be more convenient for your purposes, or whatever other reason). It makes sense to call the charts $(U,\phi)$ and $(V,\psi)$ smoothly compatible if, in a small neighborhood $O_x$ around any point $x$ that lies in both domains, we can smoothly transition from the "target" $\phi(O_x)$ to the target $\psi(O_x)$, and vice versa. If there is no point in $U\cap V$ then this condition is always fulfilled (a vacuous truth). If there is one, then the transition function $\phi\circ \psi^{-1}$ and its inverse must be smooth, i.e. both must be diffeomorphisms. Note that, since $U\cap V$ is open, we can always find a neighborhood around a point in it that is fully contained in it.

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