I'm considering a topological manifold $M$ (e.g. a 2-sphere).
Every point of $M$ (in the 2-sphere case, the whole sphere excluding the pole) is homeomorphic to an open interval of $\mathbb R^N$ (for the sphere, $\mathbb R^2$). This allows to define charts.
However, on so much contributions over the web, the stressed feature is the fact that just a very small neighborhood of a point $p \subset M$ "looks like" a flat portion of space (the usual example from cartography, see e.g. the first answer in this discussion), which seems me misleading.
Where am I wrong?
The property of 'looking flat' is weaker than what we intuitively think it is. In the example of cartography, the time when we get into trouble is when we try to map the whole earth at once. 'Small' maps where at least one point is taken away are contorted, but we understand what they mean. Most importantly, we can jump between such maps (turning the page in an atlas) in a homeomorphic fashion.
When you are trying to convince people of this with differential manofolds, it's much easier to talk about very small map snippets, because at that scale the then differentiable functions are almost identity maps. It not only can be mapped to a flat surface, it already 'looks flat'.