The category of topological spaces has a forgetful functor to set which commutes with both small limits and colimits (it has both a left and a right adjoint). Moreover Set is a Grothendieck topos and in any Grothendieck topos colimits are stable under pull-backs in the sense that $$ colim_\alpha (X_\alpha \times_Y Z) \cong (colim_\alpha X_\alpha) \times_Y Z.$$

I'm not an expert, but I don't think the topological spaces form a Grothendieck topos. I'd be happy to learn the contrary. But this raises the question: Does this formula hold in topological spaces? Notice that because this equation holds in Set this is just a question about two topologies on the same set.

We could also ask some related questions:

- If this doesn't hold generally, then does it hold if we assume the map $Z \to Y$ is
*open*? - What if $Z \to Y$ and all the maps $X_\alpha \to Y$ are open?
- What if we assume something nice about our spaces? For example suppose they are sober? or compactly generated? or Hausdorff?