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We consider the analogous statement (which we call Hurewicz dichotomy) for $\\Sigma^1_1$ subsets of the generalized Baire space ${}^\\kappa \\kappa$ for a given uncountable cardinal $\\kappa$ with $\\kappa=\\kappa^{<\\kappa}$, and show how to force it to be true in a cardinal and cofinality preserving extension of the ground model. 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