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We prove that every (extended) Borel subset E of X x Y, where X is a complete metric and Y is Polish, can be covered by countably many extended Borel graphs of mappings from X to Y if the sections E x = {y 6 Y: (x, y) ∈ E}, x ∈ X, are countable. This is a nonseparable version of a classical theorem of Luzin and Novikov.

Borel sets with countable sections for nonseparable spaces