Hereditary normality of γN

One of the classical separation axioms of topology is complete normality. A topological space X is completely normal if for every pair of subsets A and B of X which are separated (i.e. A ̄ ∩ B = ∅ = A ∩ B̄) there are disjoint open sets containing A and B respectively. A standard exercise is to show that this is equivalent to hereditary normality; that is, the property that all subspaces of X are normal. Hausdorff spaces satisfying this property are commonly designated as T5 spaces. Until now it has been somewhat of a mystery how well-behaved countably compact T5 spaces can be. Under Gödel’s axiom of constructibility (V = L) they can be quite pathological: in [Fe] there is a V = L construction of a compact T5 space X of cardinality 2 c with a countable dense subspace, yet X has no nontrivial convergent sequences. Until the results of this paper, some of which were announced in [NSSV], it was not known whether such a space can be constructed just using the usual axioms of set theory [ZFC]