# Example of a space which is Hausdorff and locally Euclidean but not paracompact?

On page 330 of John G. Ratcliffe's text on hyperbolic manifolds, he defines a manifold $M$ to be a locally Euclidean Hausdorff space. By locally Euclidean, he means as usual that for each $x \in M$ there is an open neighbour $U$ of $x$ which is homeomorphic to an open subset of $\mathbb{R}^n$.

But he omits the usual paracompactness assumption. I would guess that $M$ being locally Euclidean and Hausdorff is not sufficient to imply that it is paracompact, given this problem from Lee's book.

Does anyone know of such a counterexample?

## 2 Answers

There's the long line. $\qquad$

• Thanks! Looks like that does the trick. – ಠ_ಠ Jun 10 '17 at 12:51

Note that every metrizable space is paracompact and $\mathbb R^n$ with usual topology is metrizable.

• Also, a paracompact Hausdorff locally metrizable space is metrizable. So for manifolds, paracompactness and metrizable are equivalent. – Henno Brandsma Jun 10 '17 at 14:43