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I need to clear up my confusion on the definition of a smooth manifold. So we say that $M$ is a smooth manifold (of dimension $n$), if $M$ is Hausdorff and if every $x \in M$ is contained in a neighborhood $U$ that's homeomorphic to an n-ball (the pair $\phi, U$ is called a chart), and if two such charts $\phi_1, U_1$, and $\phi_2, U_2$ overlap, then

$$\phi_2 \circ \phi_1^{-1} : \phi_1(U_1 \cap U_2) \to \phi_2(U_1 \cap U_2)$$

is a smooth map.

But I remember my professor proving that a certain space was a smooth manifold by merely finding an atlas (an open covering of the space by charts) such that the above holds. But according to the definition I wrote, this would be insufficient. Can anyone clear up my confusion?

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    Why would that be insufficient?2012-10-13
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    Well, consider two atlases $A1$ and $A2$. It could be that all transition maps within $A1$ and within $A2$ are smooth, but if two charts (one from A1 and one from A2) overlap, then that transition map may not be smooth, and the definition I wrote(which may be wrong), indicates that any transition map must be smooth.2012-10-13
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    A manifold is a pairing of a topological space with an atlas. Topological spaces can support many incompatible manifold structures. There is not always a canonical choice. Your professor has taken a topological space and given it a smooth structure, so you get a manifold. Whether or not it is compatible with some other atlas doesn't really matter.2012-10-13
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    My preferred definition of a smooth manifold is that of using sheaves. We consider a ringed space $(M, \mathcal{O}_M)$, where $M$ is a Hausdorff space and $\mathcal{O}_M$ is a subsheaf of the sheaf of germs of continuous functions on $M$. Let $V$ be an open subset of $\mathbb{R}^n$. Let $\mathcal{O}_V$ be the sheaf of germs of smooth functions on $V$. The ringed space $(V, \mathcal{O}_V)$ is called a model space. If every point $p$ of $M$ has an open neighborhood $U$ such that $(U, \mathcal{O}_M|U)$ is isomorphic to a model space, $(M, \mathcal{O}_M)$ is called a smooth manifold.2012-10-13

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