In this paper, we obtain a generalization of Grothendieck's theorem for the space of continuous mappings C & lambda;,& mu;(X, Y ) where Y is a complete uniform space with the uniformity & mu; endowed with the topology of uniform convergence on the family & lambda; of subsets of X. A new topological game is defined - the Asanov-Velichko game, which makes it possible to single out a class of topological spaces of the Grothendieck type.The developed technique is used to generalize the Grothendieck's theorem for the space of continuous mappings endowed with the set-open topology.& COPY; 2023 Elsevier B.V. All rights reserved.