Weakly normal topological spaces and products

The behavior of the property of weak normality with respect to topological products is examined versus normality. The following generalization of Tamanoʹs theorem is proved: if X × βX is weakly normal then X is paracompact. Some versions of Katětovʹs theorem are obtained. In particular, it is proved that if X × Y is hereditarily weakly normal then either each countable subset of X is closed or each convergent free sequence in Y has countable cofinality.