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.
We will show that, consistently, every uncountable set can be continuously mapped onto a non measure zero set, while there exists an uncountable set whose all continuous images into a Polish space are meager.2000 Mathematics Subject Classification. 03E17.
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