I want to prove
A manifold $M$ can be covered by a countable collection of neighbourhoods each diffeomorphic to an open subset of $\mathbb R^m$.
By definition we have for each $x \in M$ an open neighbourhood diffeomorphic to an open subset of $\mathbb R^m$. But I don't understand why they have to be countable. Here a manifold is subset of Euclidean space.
Manifold is a subset $M$ of $\mathbb R^n$ is a $k$ - dimensional manifold if $\forall x\in M $ there is open $U,V \subset \mathbb R^n$ $x \in U$ and a diffeomorphism $f:U\rightarrow V$ such that $f(U \cap M)=V \cap(\mathbb R^k \times 0) $.