We often assume manifolds to be paracompact Hausdorff. Clearly, this implies normal.


However, there is a manifold (I mean locally Euclidean Hausdorff space) which is not paracompact. Without paracompactness, they are still regular because manifolds are locally compact, but does it imply normal?


The only example of non paracompact manifold which I know is the "long line". However, I hear this is normal. I want to know whether manifold implies normal or not.



"Abnormal" manifold
The fiberwise one-point-compactification of the tangent bundle of the long line (pick any smooth structure) is not normal.


The two distinguished sections of this $S^1$ bundle (the zero section & the section at infinity) cannot be separated by open subsets.


