Closed-constructible functions are piecewise closed

摘要

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.We prove that if a continuous function f : X→Y between separable tnetrizable spaces maps closed subsets of X onto constructible sets of V, then X admits a countable closed cover C such that for each C ∈ C the restriction f|C is closed.