TITLE: Why do we use only compatible charts in the Theory of Manifolds?
QUESTION [5 upvotes]: I couldn't find a duplicate, although I think is a very common question.

Given two charts, ($U_{1},φ_{1}$), ($U_{2},φ_{2}$), on a n-dimensional topological manifold M, such that: $U_{1} \cap U_{2}\neq \emptyset$, we get transition maps:
$φ_{1}\circ φ_{2}^{-1} : φ_{2}(U_{1}\cap U_{2}) \rightarrow φ_{1}(U_{1}\cap U_{2})$, and
$φ_{2}\circ φ_{1}^{-1} : φ_{1}(U_{1}\cap U_{2}) \rightarrow φ_{2}(U_{1}\cap U_{2})$
Two charts, as above, are called compatible if the transition maps, as above, are homeomorphisms. If $U_{1} \cap U_{2} = \emptyset$, then they are compatible.

My question is, why do we need this behavior? In addition, if we want to define C$^{\infty}$-compatible charts, why do we need to take transition maps to be smooth, Euclidian mappings?

REPLY [5 votes]: Yes, each individual chart $\phi \colon U \to \mathbb{R}^n$ is $C^\infty$, but remember that each chart in only defined on one open neighbourhood $U \subset M$. To allow us to consider behaviour on the whole of the manifold, it's important to be able to patch the $U$ neighbourhoods together nicely, going from one chart $\phi$ to another without having any non-differentiable issues. This is why we need transition maps to be smooth as well as the charts.