T0 space

A T0 space is a topological space $X$ such that for any distinct points $x\neq y$ in $X$, there exists an open set $U$ with either $x\in U$ and $y\notin U$, or $y\in U$ and $x\notin U$.

This is the weakest of the common separation axioms; it is implied by being T1 , and hence by being Hausdorff .

Examples:

• The Sierpiński space on $\{0,1\}$ with open sets $\varnothing$, $\{1\}$, and $\{0,1\}$ is T0 but not T1.
• Any metric space is T0 (in fact, it is Hausdorff).