Hereditary topological diagonalizations and the Menger-Hurewicz conjectures

We consider the question of 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 are provably hereditary. This is in contrast with the Borel case, where some of the classes are provably hereditary. Two of the examples are counter-examples of sizes 8 and b, respectively, to the Menger and Hurewicz Conjectures, and one of them answers a question of Steprans on perfectly meager sets.