A manifold is usually defined as a [second-countable](http://en.wikipedia.org/wiki/Second-countable_space) [hausdorff](http://en.wikipedia.org/wiki/Hausdorff_space) 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](http://en.wikipedia.org/wiki/Paracompact_space), 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](http://en.wikipedia.org/wiki/Long_line_%28topology%29) paracompact? Should I consider it a manifold?



Is the long line paracompact?
Here's another proof, which shows that any connected paracompact locally Euclidean space X is second-countable. Cover X by Euclidean charts and take a locally finite refinement. Say an open set is good if it only intersects finitely many of the charts. Now take any point x and take a good neighborhood of it. The charts that intersect that good neighborhood can then themselves be covered by countably many good open sets. There are then only countably many charts intersecting those good open sets, and those charts can be covered by countably many good sets. Iterating this countably many times, you get an open set U associated to x which is covered by countably many charts such that if a chart intersects U, it is contained in U. It follows that the complement of U is also a union of charts, so by connectedness U is all of X. Thus X can be covered by countably many charts and is second-countable.


