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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

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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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    @t.b. thanks. Obviously the union of two orientable manifolds is orientable2012-08-15
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    Of course, hence "quibble" :) For the sake of completeness: A detailed proof of the classification of $1$-manifolds from first principles is given in an appendix to Milnor's *[Topology from the differentiable viewpoint](http://books.google.com/books?id=BaQYYJp84cYC)*.2012-08-15
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    I was looking for a direct proof but I will see if I can adapt.2012-08-15
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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