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 a quotient map $X\to H(X)$ with the universal property that any continuous map from $X$ to a Hausdorff space factors uniquely through the Hausdorffization.

References

• The Hausdorff Quotient by Bart Van Munster