TITLE: When (why) did we allow manifolds to be non-Hausdorff and/or non-second countable?
QUESTION [17 upvotes]: I was reading David Carchedi's answer for a question on Grothendieck topology for a non-small category. It "reads" like people "choose" if they allow manifolds to be Hausdorff and/or second countable. When I came across the notion of smooth manifolds for the first time, by definition, smooth manifolds are Hausdorff and second countable.

When (why) did we allow manifolds to be  non-Hausdorff and/or non-second countable?

I have observed this when I am reading about stacks.
Is it because of the constructions when we do in the set up of manifolds (stacks) which resulted in spaces that are "same as" manifolds but are not Hausdorff or not second countable? 
Is it because of the influence of algebraic geometry?

REPLY [8 votes]: If you're asking a historical question, it is probably because of Whitney, who set out a clear account of differentiable manifolds and proved the embedding theorem, that the Hausdorff and second countability assumptions have been regarded as "standard." Nevertheless, violations of these assumptions had been contemplated prior to Whitney's work, notably the non-second-countable Prüfer surface, which dates back to the 1920s.
In addition to the examples mentioned by others, note that people studying general relativity have sometimes considered non-Hausdorff spacetime manifolds (see here for a non-paywalled version). The consensus, however, seems to be that these are mathematical curiosities that are not physically significant.  Luc and Placek amusingly quote Penrose as saying, "I must … return firmly to sanity by repeating three times: spacetime is a Hausdorff differentiable manifold; spacetime is a Hausdorff …"