TITLE: Is the long line paracompact?
QUESTION [10 upvotes]: A manifold is usually defined as a second-countable hausdorff topological space which is locally homeomorphic to Rn. 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?

REPLY [3 votes]: Let me give another proof that the long line is not paracompact. A theorem of Tamano states that a completely regular space $X$ is paracompact if and only if $X\times\beta X$ is normal where $\beta X$ denotes the Stone-Cech compactification of $X$.
Suppose that $L$ is the long line. To make our lives easier, let $X=L\cup\{0\}$ be the totally ordered set with least element $0$. Then $X$ is a manifold with boundary $0$. Then $\beta X=X\cup\{\infty\}$ where $\infty\not\in X$ and $X\cup\{\infty\}$ is given the one-point compactification topology.
Then $X\times\beta X=X\times(X\cup\{\infty\})$. Let $C=\{(x,x)|x\in X\}$ and let $D=\{(x,\infty)|x\in X\}$. Then by an easy application of Fodor's lemma, it can be shown that if $U,V$ are open sets with $C\subseteq U,D\subseteq V$, then $U\cap V\neq\emptyset$. Therefore $X\times\beta X$ is not normal, so $X$ is not paracompact.
Before I go, I should mention that the result and its proof stating that every paracompact connected manifold is second countable follows from the more general fact that a locally compact space is paracompact if and only if it is the disjoint union of open $\sigma$-compact sets. In particular, every locally compact connected space is paracompact if and only if it is $\sigma$-compact.