Аннотация:We prove that a Tychonoff space X is (sequentially) Ascoli iff for every compactspace K (resp., for a convergent sequence s), each separately continuous k-continuousfunction : X × K → R is continuous. We apply these characterizations to showthat 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 spacesand 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 isa 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 → R is continuous.
