Let $f : X \to Y$ and $g : Y \to Z$ be continuous maps (between topological spaces). Assume these hypotheses:

- $f : X \to Y$ is a split surjection, i.e. has a section.
- $g \circ f : X \to Z$ is a local homeomorphism, i.e. there is an open cover $\{ U_i : i \in I \}$ of $X$ such that, for each $i \in I$, the composite $U_i \to X \to Y \to Z$ is an open embedding.

Does it follow that $g : Y \to Z$ is a local homeomorphism?

Here are some observations:

- The question with "open map" instead of "local homeomorphism" has a positive answer. In particular, under the above hypotheses, $g : Y \to Z$ must be an open map.
- Moreover, the fibres of $g : Y \to Z$ must be discrete. So we have an open map with discrete fibres – is such a thing necessarily a local homeomorphism?
- If $f : X \to Y$ is an open map, then $g : Y \to Z$ is a local homeomorphism. Conversely, if $g : Y \to Z$ is a local homeomorphism, then $f : X \to Y$ is also a local homeomorphism (hence an open map
*a fortiori*).