T1 space

A T1 space is a topological space $X$ such that for every point $x\in X$, the singleton $\{x\}$ is a closed set in $X$. Equivalently, for any distinct points $x\neq y$ there exists an open set containing $x$ but not $y$, and (symmetrically) an open set containing $y$ but not $x$.

The T1 axiom strengthens T0 and is implied by the Hausdorff condition.

Examples:

• Every metric space is T1.
• On an infinite set $X$, the cofinite topology (open sets are $\varnothing$ and complements of finite sets) is T1 but not Hausdorff.