Finite union of H-sets and countable compact sets
In , D. E. Grow and M. Insall construct a countable compact set which is not the union of two H-sets. We make precise this result in two directions, proving such a set may be, but need not be, a finite union of H-sets. Descriptive set theory tools like Cantor-Bendixson ranks are used; they are developed in the book of A. S. Kechris and A. Louveau . Two proofs are presented; the first one is elementary while the second one is more general and useful. Using the last one I prove in my thesis, directed by A. Louveau, the existence of a countable compact set which is not a finite union of Dirichlet sets. This result, quoted in , is weaker because all Dirichlet sets belong to H. Other new results about the class H and similar classes of thin sets can be found in ,  and .