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Given a surjective mapping f : E → F between Banach spaces, we investigate the existence of a subspace G of E, with the same density character as F, such that the restriction of f to G remains surjective. We obtain a positive answer whenever f is continuous and uniformly open. In the smooth case, we deduce a positive answer when f is a C^1-smooth surjection whose set of critical values is countable. Finally we show that, when f takes values in the Euclidean space R^n, in order to obtain thisarXiv:1607.01725v2 fatcat:z2z76rmt3ncqjjwwvlfdyu7dse