What quotient space of a non Hausdorff space can be a Hausdorff space?

Question

A line with two Origins is a simple example of a quotient space of Hausdorff space that is not Hausdorff. What quotient of a non Hausdorff space (or maybe non Kolmogorov) can be a Hausdorff(or Kolmogorov) space?

Answer 1

To give a specific example relating to the one that you mentioned yourself: Let $X$ be the line with doubled origin (not Haudorff) and denote by $O_1$ and $O_2$ the two origins in $X$. Let $\sim$ be the equivalence relation generated by stipulating $O_1 \sim O_2$, that is: $$x \sim y\qquad \iff \qquad x = y \quad \text{or} \quad x,y \in \{O_1, O_2\}.$$ Then $X / {\sim}$ is homeomorphic to the ordinary affine line and, in particular, Hausdorff. In fact, the affine line is the “Hausdorffization” (mentioned by PrudiiArca) of the line with doubled orign.

In the comments, the OP asked for examples where the Hausdorffization identifies more points than just two. If $X$ is an irreducible space, then its Hausdorffization collapses to a single point. Examples of irreducible spaces include:

1. Any set with its codiscrete topology.
2. Any linear order with its Alexandrov topology.
3. An algebraic variety with its Zariski topology.

Answer 2

Take any topological space $X$ (Hausdorff or not) and consider on $X$ the equivalence relation $\sim$ such that $(\forall x,y\in X):x\sim y$. Since $X/\sim$ is a singleton, it is Hausdorff.

Answer 3

You can turn any topological space into a Hausdorff space by a process called hausdorffization. Basically you take the quotient by the minimal equivalence relation making the quotient a Hausdorff-space. This is in fact the free construction of this sort and exhibits $\mathsf{Haus}$ as reflective subcategory of $\mathsf{Top}$. In particular every map from $X$ to some Hausdorff-space $Y$ factors uniquely over $X_\text{Haus}$.