Hausdorffization

Definition

Let $X$ be a topological space. For $a,b\in X$, define $a\sim b$ if any open set containing $a$ intersects any open set containing $b$. The Hausdorffization of $X$ is the quotient map of $X$ by the equivalence relation generated by $\sim$. The term Hausdorffization is also sometimes used for the quotient space obtained after taking the map. Clearly, this is a Hausdorff space

The Hausdorffization has the following universal property: any map from $X$ to a Hausdorff space factors through the Hausdorffization of $X$.