Generalization of the Grothendieck's theorem

Abstract

In this paper, we have obtained a generalization of the Grothendieck's theorem for the space of continuous mappings $C_{λ,μ}(X,Y)$ where $Y$ is a complete uniform space with the uniformity $μ$ endowed with the topology of uniform convergence on the family $λ$ 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.