We prove that a Tychonoff space X is (sequentially) Ascoli iff for every compact space K (resp., for a convergent sequence \(\textbf{s}\) ), each separately continuous k-continuous function \(\Phi :X\hspace{1.111pt}{\times }\hspace{1.111pt}K\rightarrow \mathbb {R}\) is continuous. We apply these characterizations to show that an open subspace of a (sequentially) Ascoli space is (sequentially) Ascoli, and that the \(\mu \) -completion and the Dieudonné completion of a (sequentially) Ascoli space are (sequentially) Ascoli. We give also cover-type characterizations of Ascoli spaces and suggest an easy method of construction of pseudocompact Ascoli spaces which are not \(k_\mathbb {R}\) -spaces and show that each space X can be closely embedded into such a space. Using a different method we prove Hušek’s theorem: a Tychonoff space Y is a locally pseudocompact \(k_\mathbb {R}\) -space iff \(X\hspace{1.111pt}{\times }\hspace{1.111pt}Y\) is a \(k_\mathbb {R}\) -space for each \(k_\mathbb {R}\) -space X. It is proved that X is an \(s_\mathbb {R}\) -space iff for every locally compact sequential space K, each s-continuous function \(f:X\hspace{1.111pt}{\times }\hspace{1.111pt}K\rightarrow \mathbb {R}\) is continuous.
