# Refining monotone-light factorizations

Let $$f:X\to Y$$ be a continuous map between topological spaces. Consider the quotient map $$\pi:X\twoheadrightarrow X/E$$ given by decomposing the fibers of $$f$$ to their connected components.

In Lemma 6.2.21 his book, Engelking proves that when $$f:X\to Y$$ is closed and has compact fibers, and $$X$$ is moreover Hausdorff, then $$\pi$$ is closed. The end of the proof makes crucial use of $$f$$ being closed.

On page 102 of the book Dynamic Topology by Whyburn & Duda, it is proved that if the connected components of the fibers of $$f$$ are compact, and $$X,Y$$ are Hausdorff with $$X$$ also locally compact, then $$\pi$$ is closed. To avoid assuming $$f$$ is closed, the corresponding part of the proof basically localizes to a compact subset on the base and then uses the fact any continuous map from a compact space to a Hausdorff space is closed.

It somehow feels possible to remove the assumptions on $$X,Y$$ and replace them by assumptions on $$f$$ without assuming $$f$$ is closed and still have $$\pi$$ be closed.

Definition. Say a continuous map $$f:X\to Y$$ is locally (universally) closed if for any open neighborhood $$x\in W\subset X$$, there exist neighborhoods $$x\in U\subset W\subset X$$ and $$fy\in V\subset Y$$ such that $$fU\subset V$$ and also the induced map $$U\to V$$ is (universally) closed.

Question. Suppose $$f$$ is separated and locally universally closed (suppose also compact fibers if necessary). Is $$\pi$$ closed?

The separation arguments in the proofs of Engelking and Whyburn seem to carry through when merely assuming $$f$$ is separated, so it really seems the only challenge is to remove the global assumption of $$f$$ being closed or the global assumption on the source and target spaces.

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