Functions on products X Y with applications to Ascoli spaces, kₑ-spaces and sₑ-spaces

We prove that a Tychonoff space X is (sequentially) Ascoli iff for every compact space K (resp. , for a convergent sequence s), each separately continuous k-continuous function Φ: X K R is continuous. We apply these characterizations to show that an open subspace of a (sequentially) Ascoli space is (sequentially) Ascoli, and that the μ-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 kR-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 kR-space iff X Y is a kR-space for each kR-space X. It is proved that X is an sR-space iff for every locally compact sequential space K, each s-continuous function f: X K is continuous.