  We consider the question, which of the major classes defined by topological
diagonalizations of open or Borel covers is hereditary. Many of the classes in
the open case are not hereditary already in ZFC, and none of them is provably
hereditary. This is contrasted with the Borel case, where some of the classes
are provably hereditary. Two of the examples are counter-examples of sizes d$
and b, respectively, to the Menger and Hurewicz Conjectures, and one of them
answers a question of Steprans on perfectly meager sets.
