HEREDITARY NORMALITY OF GAMMA-N-SPACES

A gamma N-space is a locally compact Hausdorff space with a countable dense set of isolated points, and the rest of the space homeomorphic to omega(1). We show that under the Open Coloring Axiom (OCA) no gamma N-space is hereditarily normal. This is the key to showing that some sweeping statements are consistent with (and independent of) the usual axioms of set theory, including: (1) Every countably compact, hereditarily normal space is sequentially compact. (2) Every separable, hereditarily normal, countably compact space is compact and Frechet-Urysohn. (3) The arbitrary product of countably compact, hereditarily normal spaces is countably compact. Not all of these conclusions follow just from MA + CH: a forcing construction is given of a model of MA + c = kappa where kappa is any cardinal greater than or equal to N-2, satisfying kappa = 2(<kappa), and there is a hereditarily normal gamma N-space.