A CANTOR-BENDIXSON THEOREM FOR THE SPACE OMEGA-1 OMEGA-1

Consider the Baire space N = omega-omega, i.e. the product of countably infinitely many copies of the discrete space omega of natural numbers. The Cantor-Bendixson theorem says that any closed subset of omega-omega can be uniquely expressed as the disjoint union of a perfect (i.e. closed and dense-in-itself) set and a countable set. We study the generalized Baire space N1 = omega-1-omega-1 obtained from the cartesian product of omega-1 copies of omega-1 by letting basic neighborhoods of any f: omega-1 --> omega-1 be sets of the form N (f, alpha) = {g: omega-1 --> omega-1\g(beta) = f(beta) for beta < alpha} where alpha < omega-1. This is an omega-1-metrizable space in the sense of Sikorski [8]. We study the cardinality of closed subsets of this space. We show among other things that a natural generalization of the Cantor-Bendixon theorem to the space N1 is consistent relative to the consistency of a measurable cardinal.