Normality in products with a countable factor

The class of normal spaces that have normal product with every countable space is considered. A countably compact normal space X and a countable Y such that X×Y is not normal is constructed assuming CH. Also, ⋄ is used to construct a perfectly normal countably compact X and a countable Y such that X×Y is not normal. The question whether a Dowker space can have normal product with itself is considered. It is shown that if X is Dowker and contains any countable non-discrete subspace, then X2 is not normal. It follows that a product of a Dowker space and a countable space is normal if and only if the countable space is discrete. If X is Rudin's ZFC Dowker space, then X2 is normal. An example of a Dowker space of cardinality ℵ2 with normal square is constructed assuming .