Hereditary normality of γN-spaces

A γN-space is a locally compact Hausdorff space with a countable dense set of isolated points, and the rest of the space homeomorphic to ω1. We show that under the Open Coloring Axiom (OCA) no γN-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. ery countably compact, hereditarily normal space is sequentially compact. ery separable, hereditarily normal, countably compact space is compact and Fréchet-Urysohn. e arbitrary product of countably compact, hereditarily normal spaces is countably compact. ll of these conclusions follow just from MA + ¬ CH: a forcing construction is given of a model of MA + c = κ where κ is any cardinal ⩾ ℵ2 satisfying κ = 2<κ, and there is a hereditarily normal γN-space.