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How to prove that every $1$-manifold is orientable?

Can I use Zorn's Lemma and produce a maximal orientable manifold that will have to be all M?

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    Wouldn't it be easiest just to list them?2012-08-15

1 Answers 1

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There are two connected 1-dimensional manifolds. The circle and the real line. Both are obviously orientable because the volume forms $d\theta$ and $dx$ are non-vanishing.

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    You can start with a chart, and see if you can extend $\frac{d}{dx}$ to a vector field in the whole manifold.2012-08-15