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If C and D are irreducible, affine varieties over an algebraically closed field, and I form the product variety CxD, is the projection morphism from CxD to C necessarily an open map? That is, is the projection of each Zariski open subset of CxD necessarily Zariski open in C?

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Yes. More generally, flat morphisms locally of finite presentation are universally open (EGA IV2, Théorème 2.4.6).

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