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From this topic : Definition of Maximal atlas I found the definition of a Maximal Atlas.

But I don't understand how a maximal atlas could exist.

Indeed, I understand the maximal atlas as the one such as if we add another map then it will no longer be compatible.

But if we take $\mathbb{R}^2$ as manifold and imagine that we have a chart $(U,\phi_U)$ that maps the subset $U$ of $\mathbb{R}^2$ to a disk in $\mathbb{R^2}$.

For example, we take $U$ the disk of radius $1$ and $\phi_U$ the identity ( $\phi_U:x\in U \mapsto x\in \mathbb{R}^2$).

We can take $V \subset U$ a disk of radius $1/2$ and $\phi_V(x)=x+(1,0)$ such as $(V,\phi_V)$ is another chart. Thoose two charts are compatible (I can find a map from the disk centered at the origin and the disk centered in $(1,0)$).

And if we iterate we can always take a subset inside the previous one and we can add as much charts as we want and they would be compatible.

Here I took $M=\mathbb{R}^2$ as the manifold of example but this could be generalized to any M.

Where do I miss something ?

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Your argument demonstrates that a maximal atlas must contain a lot of charts. It doesn't negate its existence.

In the particular case of $\mathbb{R}^2$, the usual smooth structure (maximal smooth atlas) is the set of all charts that are smoothly compatible to the identity map $i:\mathbb{R}^2\to\mathbb{R}^2$. In particular, this will contain $(U,\phi_U)$, $(V,\phi_V)$, and all other such charts.

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  • $\begingroup$ So in fact a maximal atlas can contain an infinite number of charts ? (Like in my example) $\endgroup$ – StarBucK Aug 21 '17 at 1:07
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    $\begingroup$ @StarBucK Yes! It will certainly contain infinitely many. $\endgroup$ – John Griffin Aug 21 '17 at 1:08
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    $\begingroup$ Indeed, uncountably many. $\endgroup$ – Ted Shifrin Aug 21 '17 at 5:17

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