Suppose we have a topological manifold $M$ so by definition we have that,
(1) $M$ is Hausdorff
(2) $M$ is second countable
(3) $M$ is locally euclidean
I have a question but I don't know if it is the right question or the wrong question
so question is since $M$ is locally euclidean that is for all $p\in M$ there exists an open set $p\in U$ and a map $\phi:U\to \mathbb{R^n}$ which is homeomorphism.And then $n$ will be the dimension of $M$ and $(U,\phi)$ as a chart, can we say anything about for n-dimensional manifold we need a minimum $m$ number of charts to cover the whole manifold.Is this a valid question? or there can be a manifolds which always need infinite number of charts?