A manifold is usually defined as a second-countable hausdorff topological space which is locally homeomorphic to **R**^{n}. My understanding is that the reason "second-countable" is part of the definition is to make sure that the space is paracompact, which you want so that you get locally finite partitions of unity. Once you have locally finite partitions of unity, basically anything you can multiply by a function can be constructed locally (any presheaf that is a module over the sheaf of functions is automatically a sheaf), a property you want manifolds to have.

But do we unnecessarily throw out some paracompact topological spaces which "should" be manifolds by requiring second-countability. A boring example is an uncountable disjoint union of manifolds, but there are other more interesting spaces that kind of look like they should be manifolds.

In particular, is the long line paracompact? Should I consider it a manifold?