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Given diagrams of topological spaces $$X_0\rightarrow X_1\rightarrow\ldots$$ $$Y_0\rightarrow Y_1\rightarrow\ldots$$ $$Z_0\rightarrow Z_1\rightarrow\ldots$$ and maps $X_i\rightarrow Z_i$, $Y_i\rightarrow Z_i$, such that everything commutes. Taking the colimit yields a canonical map $$colim(X_i\times_{Z_i}Y_i)\rightarrow colim(X_i)\times_{colim(Z_i)}colim(Y_i)$$ from the colimit of the pullbacks to the pullback of the colimits. This map is a continuous bijection, since filtered colimits commute with finite limits in the category of sets and the forgetful functor from spaces to sets preserves all limits and colimits.

This answer of @StefanHamcke serves as a counterexample where the map is not a homeomorphism even in the case where the first three diagrams consist of closed inclusions.

I'm interested in mild point set topological restrictions on the spaces and maps, such that the canonical map is a homeomorphism. The answer linked suggests, that separation conditions will be necessary, so what e.g. about closed inclusions of T1 spaces, Hausdorff spaces, etc.?

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