2. Separation conditions

Here are some separation conditions of increasing strength. Which (if any) is needed depends on the application.

$T_0$ space: The open sets distinguish points. In other words, given two points in $X$, there is an open set that contains one point but not the other.

$T_1$ space: The open sets distinguish points both ways around: Given two points, for each point there is an open set that contains it but not the other point. An equivalent condition is that every singleton is a closed set.

$T_2$ space, or Hausdorff space: Any two points can be separated by disjoint open sets. In other words, given two points $x, y$, there are disjoint open sets $O _ x, O _ y$ with $x \in O _ x$ and $y \in O _ y$.

There are stronger possibilities as well.