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I am trying to prove that covering of a manifold is again manifold.

I was wondering if the next holds: When considering manifolds,

is an open subset of "Euclidean neighbourhood" again Euclidean nbh?

Remark: where by E. nbh, I mean an open set on manifold which is homeomorphc to Euclidean space.

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    Homeomorphic to $\mathbb{R}^n$ in general no. The open subset need not be connected. If connected, it can have "holes" if $n > 1$ (think of a spherical shell for a simple example). Homeomorphic to an open subset of $\mathbb{R}^n$, yes. – 2017-02-13
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    @DanielFischer I was wondering if for Euclidean nbh. I can take those trivialising neighbourhoods from the definition of a based space for coverings? – 2017-02-13

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