Usually manifolds tout court are required to be second countable.
– Mariano Suárez-Alvarez♦May 15 '11 at 20:42

2

Sorry, I should state this question more carefully. Of course, Zev Chonoles and Mariano Suarez-Alvarez are right: the usual definition of a manifold requires second-countability and Hausdorff and locally euclidean. My question should merely be: At which point in the theory of Lie groupoids does one really need that the base manifold is second-countable? When constructing a Lie groupoid from a foliation one actually has to be a bit careful at this point. If one takes uncountably many charts the base manifold of the Lie groupoid won't be second-countable.
– Dave LewisMay 15 '11 at 20:57

2

@Dave Lewis: Can I request that you edit your question to include your comments above? (Mark the edit as an edit, so that @Zev and @Mariano 's comments still make sense.) It sounds like you have a more specific direction that you're thinking about, and in any case clearly recognize that "When constructing a Lie groupoid from a foliation one actually has to be a bit careful at this point", for example. I do not know of a good reason to have questions on MO that are only one sentence long, and there are many good reasons for including a few paragraphs.
– Theo Johnson-FreydMay 15 '11 at 22:50

2

@Zev, Mariano, and Dave: If you require manifolds to be second countable, then a disjoint union of manifolds is not always a manifold. Replacing second countability by paracompactness allows you to keep all good properties of second countable manifolds and makes the category of manifolds closed under coproducts, which seems like a good property to have.
– Dmitri PavlovMay 16 '11 at 4:19

Answer #1: There is no real reason for imposing that the base manifold of a groupoid be second countable.

Answer #2:
You lose some desirable properties if you don't impose second countability:
For example, without it,
the homotopy type of the geometric realisation of the nerve
will no longer be an invariant of the Morita equivalence class of the groupoid.

Re Answer #2: Weird! I would have expected that the homotopy type of the nerve was a well-defined invariant for any topological groupoid, and that the construction should factor through forgetting from Manifolds to Homotopy Types. Could you either explain more, or include a reference?
– Theo Johnson-FreydMay 15 '11 at 22:52

@Theo: I take my favourite non-second countable manifold: the long line $L$, and I look at the cover consisting of all of its bounded connected open subsets. The corresponding Cech groupoid is Morita equivalent to $L$. There is an obvious projection from the geometric realization of the Cech groupoid back to $L$. But there is no section of that map: that's because the cover does not admit partitions of unity. More generally, you can show that the projection does not admit a homotopy inverse.
– André HenriquesMay 15 '11 at 23:12

Ah, Andre - that is why you redefine Morita equivalence not to use 'local sections', but 'local sections wrt a numerable cover'. This class of weak equivalences of topological/Lie groupoids (take your pick) is closer to what people think of when they restrict to paracompact spaces.
– David RobertsMay 16 '11 at 0:35

tout courtare required to be second countable. – Mariano Suárez-Alvarez♦ May 15 '11 at 20:42