The line with two origins
Take two copies of the real line and glue them together at every point except $0$, so that each nonzero point $x$ of one copy is identified with the point $x$ of the other. The result looks like $\reals$, except that it has two origins, $0_1$ and $0_2$: every neighborhood of $0_1$ and every neighborhood of $0_2$ contain the same small nonzero numbers.
Terms
| Term | This example | Why |
|---|---|---|
| T1 space | ✓ | The line with two origins is T₁, because each point is closed. |
| hausdorff space | ✗ | |
| normal space | ✗ | |
| metrizable | ✗ | The line with two origins is not Hausdorff, and so not normal or metrizable, because the two origins cannot be separated by disjoint open sets; this is why manifolds are required to be Hausdorff. |