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The empty set can be regarded as an object in the category of smooth manifolds, at least for technical considerations.

Is the empty set an orientable manifold?

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    Just a note: The authors of this book http://books.google.no/books?id=i3FYIWIYu5QC&pg=PA327&lpg=PA327&dq=%22empty%20set%22%20orientable&source=bl&ots=lp_k2_LYO5&sig=4FlBGXlBsUr5ka974taa7mpdRXY&hl=en&sa=X&ei=g-6LUOPnOZGM4gSnp4HoCg&redir_esc=y#v=onepage&q=%22empty%20set%22%20orientable&f=false stipulate in the footnote on page 327 that the empty set *is* orientable.2012-10-27

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EDIT: This is wrong. See Henning Makholm's answer below.

An $n$-manifold is orientable if and only if it has a nonzero differential form taking $n$ arguments. The empty manifold has no nonzero forms whatsoever, so it is not orientable in any dimension.

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    The relevant condition is not at the differential form must be different from the everywhere zero form, but that _at each point in the manifold_, the differential form's restriction to that point's tangent space must be nonzero. This is true of the empty map, because there are no points to check it for.2012-10-27
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Contradicting Espen's answer: The empty map does (vacuously) provide a nonzero differential form at every point on the manifold, and does so continuously. Therefore the empty manifold is orientable in every dimension.

(But really this will depend on the exact definition you're working with, and among the various usually-assumed-equivalent definitions for "orientable" there are some that are only really equivalent when the manifold is assumed to be nonempty).

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    But one often sees orientability defined in terms of such theorems... edit: to be clear, which definition of orientability should we apply here?2012-10-27