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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? –  Sean Eberhard Aug 15 '12 at 8:54

1 Answer 1

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 orientable –  Thomas Rot Aug 15 '12 at 9:01
    
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. –  t.b. Aug 15 '12 at 9:08
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I was looking for a direct proof but I will see if I can adapt. –  André Lima Aug 15 '12 at 9:22
    
You can start with a chart, and see if you can extend $\frac{d}{dx}$ to a vector field in the whole manifold. –  Thomas Rot Aug 15 '12 at 9:32

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