T0 space

Definition

A topological space $X$ is T0 when for all $x \neq y$ in $X$, some open set contains exactly one of $x$ and $y$.

Remarks

Each T1 space is T0. The Sierpiński space $\{0, 1\}$, with open sets $\varnothing$, $\{1\}$ and $\{0, 1\}$, is T0 but not T1. An indiscrete space with at least two points is not T0.

Explanations

L1 human

In a T0 space, no two distinct points have exactly the same open sets around them. The open sets may still fail to separate the points in both directions.