TITLE: Do pushouts of compactly generated Hausdorff spaces exist?
QUESTION [5 upvotes]: Let $A\to X$ and $A\to Y$ be maps of compactly generated Hausdorff spaces. Does the pushout $X\coprod_A Y$ in the category of compactly generated Hausdorff spaces exist? If necessary, one can assume that both these maps are cofibrations.

REPLY [4 votes]: Both categories $\mathbf{CG}$ and $\mathbf{Haus}$ have inclusion functors into $\mathbf{Top}$ which are part of an adjunction. While $\mathbf{CG}↪\mathbf{Top}$ is left adjoint to the functor $k$, with the counit being the identity map from the $k$-ification of $X$ to $X$ with its original topology, the inclusion $\mathbf{Haus}↪\mathbf{Top}$ is right adjoint to the functor $H$, the unit being the quotient map $X\twoheadrightarrow HX$ onto the largest Hausdorff quotient $HX$. These two categories are independent in the following sense:

The reflector $H:\mathbf{Top}\to\mathbf{Haus}$ restricts to a reflector $\mathbf{CG}\to\mathbf{CGHaus}$ since each quotient of a compactly generated spaces is again compactly generated.
The coreflector $k:\mathbf{Top}\to\mathbf{CG}$ restricts to a coreflector $k:\mathbf{Haus}\to\mathbf{CGHaus}$ since $k$ only refines the topology on a space.

The functor $H:\mathbf{CG}\to\mathbf{CGHaus}$ is a left adjoint, so it preserves colimits. That mean that for a diagram $D$ in $\mathbf{CGHaus}$ the colimit in $\mathbf{CGHaus}$ is the same as the Hausdorff quotient $H$ of the colimit in $\mathbf{CG}$, which coincides with the colimit of $D$ in $\mathbf{Top}$ as the inclusion $\mathbf{CG}↪\mathbf{Top}$ preserves colimits.
Dually, the functor $k:\mathbf{Haus}\to\mathbf{CGHaus}$ is a right adjoint, so it preserves limits. That means that for a diagram $D$ in $\mathbf{CGHaus}$ the limit in $\mathbf{CGHaus}$ is the same as the $k$-ification of the limit of $D$ in $\mathbf{Haus}$, which coincides with limit in $\mathbf{Top}$.
Since $\mathbf{Haus}$ is complete, so is $\mathbf{CGHaus}$. And since $\mathbf{CG}$ is cocomplete, so is $\mathbf{CGHaus}$.
So if you want the pushout in $\mathbf{CGHaus}$, then take the pushout in $\mathbf{Top}$ and apply the Hausdorff quotient $H$ to it.