A topological space $(X,\tau)$ is said to be $T_1$ (or said to hold the $T_1$ axiom) if for all distinct points $x,y\in X$ ($x\neq y$ ), there exists an open set $U\in\tau$ such that $x\in U$ and $y\notin U$ .

A space being $T_1$ is equivalent to the following statements:

• For every $x\in X$ , the set $\{x\}$ is closed.
• Every subset of $X$ is equal to the intersection of all the open sets that contain it.
• Distinct points are separated.