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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) $.

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    $\begingroup$ What is your definition of a manifold? $\endgroup$
    – D. Brogan
    Jul 5 '18 at 16:01
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    $\begingroup$ Presumably, you have to use second-countability of the Euclidean space your manifold is embedded into. $\endgroup$ Jul 5 '18 at 16:13
  • $\begingroup$ If we assume that then there is nothing left to prove. I just learned that manifold is also defined such that it is second countable. But the definitions (the one I provided and this) should be equivalent and hence I am now concerned with proving the equivalence at least one way. $\endgroup$ Jul 5 '18 at 16:28
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As you state that manifolds are a subspace of some $\mathbb{R}^n$, this means they are second countable, and in particular Lindelöf. This implies that the cover by "open neighbourhoods of $M$ that are diffeomorphic to an open subset of $\mathbb{R}^k$" of $M$ has a countable subcover. Done.

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