Example of colimit of Hausdorff spaces which is not Hausdorff In https://mathoverflow.net/questions/195248/co-hausdorffification, it is mentioned that the subcategory of Top consisting of Hausdorff spaces is not closed under colimits.
The simplest colimit I could think of is the disjoint union, but this doesn't appear to provide a counterexample.
What is an example of a colimit of Hausdorff spaces which is non-Hausdorff?

Quotient spaces of Hausdorff spaces may (sadly) not be Hausdorff themselves. So consider a Hausdorff space $X$ and an equivalence relation $\mathcal R\subset X\times X$ on $X$ such that the quotient is non Hausdorff. The quotient space $X/\mathcal R$ is the coequalizer of the two projections $\mathcal R\xrightarrow{\pi_1,\pi_2} X$:
$$\mathrm{Coeq}\left(\mathcal R\,\underset{{\pi_2}}{\overset{\pi_1}{\rightrightarrows}}X\right)=X/\mathcal R$$
where $\mathcal R$ is given the subspace topology from $X\times X$, which is Hausdorff. This gives an example of a colimit of Hausdorff spaces that isn't Hausdorff.
