Recall a continuous map is separated if its diagonal is closed. This is equivalent to the fibers being relatively Hausdorff in the total space.
Proposition. Suppose $\mathrm P$ is a class of continuous maps satisfying the following conditions.
- Closed under composition.
- Stable under pullback.
- Contains closed topological embeddings.
Then $g\circ f$ lies in $\mathrm P$ and $g$ is separated, then $f$ has $\mathrm P$.
Proof. Form the obvious pullback diagrams and play around.
Question. What is the topological intuition behind this proposition? How does the separation property of fibers somehow allow to "cancel" separated maps?
