  In this paper, we have obtained a generalization of the 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 theorem for the
space of continuous mappings endowed with the set-open topology.
