Grothendieck’s Theorem on the Precompactness of Subsets of Functional Spaces over Pseudocompact Spaces

Generalizations of the theorems of Eberlein and Grothendieck on the precompactness of subsets of function spaces are considered: if \(X\) is a countably compact space and \(C_p(X)\) is a space of continuous functions on \(X\) in the topology of pointwise convergence, then any countably compact subspace of the space \(C_p(X)\) is precompact, that is, it has a compact closure. The paper provides an overview of the results on this topic. It is proved that if a pseudocompact \(X\) contains a dense Lindelöf \(\Sigma\)-space, then pseudocompact subspaces of the space \(C_p(X)\) are precompact. If \(X\) is the product Čech complete spaces, then bounded subsets of the space \(C_p(X)\) are precompact. Results on the continuity of separately continuous functions are also obtained.