Examples

Hausdorff manifolds - a non-Hausdorff example from data

Hausdorff manifolds?

Consider $X=\mathbb{R}\backslash \{0\}\bigsqcup \{B\}\bigsqcup \{R\}$ with the standard non-Hausdorff metric:

So the points $B$ and $R$ cannot be separated by a disjoint neighborhood.

But this can actually happen with data! Suppose flowers come in two colors, red and blue, but the value (lightness) ranges from very dark (black) to very light (white):