A topological space $(X,\tau)$ is said to be $T_0$ (or to satisfy the $T_0$ axiom ) if for all distinct $x,y\in X$ there exists an open set $U\in\tau$ such that either $x\in U$ and $y\notin U$ or $x\notin U$ and $y\in U$ .

All $T_1$ spaces are $T_0$ . An example of $T_0$ space that is not $T_1$ is the $2$ -point Sierpinski space.