Closed-constructible functions are piecewise closed

A subset of a topological space X is constructible if it belongs to the smallest algebra of subsets that contains all open subsets of X. ve that if a continuous function f : X → Y between separable metrizable spaces maps closed subsets of X onto constructible sets of Y, then X admits a countable closed cover C such that for each C ∈ C the restriction f | C is closed.