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?
 A: 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.
A: Consider the pushout 
$$\begin{matrix}
[0,1) &\to & [0,1] \\
\downarrow & & \downarrow \\
[0,1] &\to & X
\end{matrix}$$
where all maps are inclusions. Here $X$ (an interval "with two $1$'s") is not Hausdorff.
Edit: I realized that a diagram without a colimit in $\mathbf{Haus}$ is different from a diagram whose colimit is not Hausdorff. In that MO question you mention, someone commented that Hausdorff spaces are not closed under taking colimits. The above diagram is an example where the colimit in $\mathbf{Top}$ is not Hausdorff, so it is not the colimit in $\mathbf{Haus}$. But that does not mean that this diagram of Hausdorff spaces does not have a colimit in $\mathbf{Haus}$, it is just not the usual colimit. In fact, every diagram in $\mathbf{Haus}$ has a colimit in $\mathbf{Haus}$, and it is obtained by applying the largest Hausdorff quotient functor $h:\mathbf{Top}\to\mathbf{Haus}$ to the colimit in $\mathbf{Top}$. This is because the functor $h$ preserves colimits, being left adjoint to the inclusion $i:\mathbf{Haus}\to\mathbf{Top}$. So the category $\mathbf{Haus}$ is closed  under colimits.
So if the functor $i$ preserves a colimit, then the colimit in $\mathbf{Top}$ is a Hausdorff space. Conversely, if the colimit is Hausdorff, then applying $h$ (which on Hausdorff spaces is the identity) gives just this space as the colimit in $\mathbf{Haus}$, so the inclusion $i$ preserves the colimit.
For the diagram above, the colimit in $\mathbf{Haus}$ is the space $[0,1]$ since in the Hausdorff quotient $h(X)$ both $1$'s are identified.
