Light maps and extensional dimension

We show that any light map f: X → Y between compact spaces admits a decomposition f = gh, where g : Z → Y is a finite-to-one map of a special type and h : X → Z has arbitrarily small fibers. It follows that light maps between compact spaces do not lower extensional dimension. Our theorem yields a positive answer to Problem 423 from “Open Problems in Topology”. We also generalize Hurewiczʹ theorem on dimension-raising maps to the case of extensional dimension.