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

Question

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?

score 4 · Accepted Answer · answered Jun 10 '17 at 12:49

4

There's the long line. $\qquad$

answered Jun 10 '17 at 12:49

Angina Seng

158,341

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

score 2 · Answer 2 · answered Jun 10 '17 at 12:47

2

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

answered Jun 10 '17 at 12:47

Paul

20,553

2

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

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

2 Answers2