gn.general topology - Is the long line paracompact?

For connected locally Euclidean spaces, paracompactness is equivalent to second countable. Therefore if we truly insisted that our manifolds be second countable then the only thing that we would lose would be arbitrary coproducts. However, we don't insist that. Although it's often stated in the definition early on in a differential topology course or book, in my experience that's just because it's easier to explain than paracompactness (I tend to pick metrisable, myself). Of course, that early in a course we're probably not too concerned about arbitrary coproducts either.

Moreover, because manifolds split into a coproduct of their connected components, we tend to deal with connected manifolds unless we really can't avoid it. And even when they aren't connected, they most often have countably many components. So in practise the distinction doesn't arise.

I don't have it in front of me here, but I believe that an appendix to the first volume of Spivak's "Introduction to differential geometry" contains a proof of four equivalent conditions for locally Euclidean spaces (perhaps requiring Hausdorff). If I remember aright, the conditions are: paracompact, second countable, metrisable, and σ-compact.

There was a paper on the arxiv on Monday, 0910.0885, which lists 107 conditions for a connected locally Euclidean Hausdorff space equivalent to that it be metrisable. Amongst them are paracompactness and second countable.