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If not, what is an example of a non-compact, open manifold that is non-orientable? So if non-orientability $\Rightarrow$ compactness, is there a theorem and what is the proof?

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A Möbius strip without boundary is not compact.

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  • $\begingroup$ True. I think was also asking for, assuming one exists, is an example of a non-orientable manifold that is not totally bounded. $\endgroup$ – Mr X Nov 26 '18 at 18:26
  • $\begingroup$ @MrX: I'm not sure what that means for a general topological manifold. How about gluing an infinitely long boundaryless ribbon to your Möbius strip at right angles? $\endgroup$ – Henning Makholm Nov 26 '18 at 18:33

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