Subnormality in subspaces of products

A T1-space X is called subnormal if every two disjoint closed subsets of X are contained in some disjoint Gδ-sets. We present a statement on hereditary subnormality of the product of two spaces which is an extension of Katětovʹs theorem. It is shown that the fact that all finite powers of a compact Hausdorff space X are hereditarily subnormal does not imply that X is first countable. Some sufficient conditions for hereditary subnormality of products are obtained.