Hereditary normality of γ [formula omitted]-spaces

Abstract

A γ N -space is a locally compact Hausdorff space with a countable dense set of isolated points, and the rest of the space homeomorphic to ω 1. We show that under the Open Coloring Axiom (OCA) no γ 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. (1) Every countably compact, hereditarily normal space is sequentially compact. 2. (2) Every separable, hereditarily normal, countably compact space is compact and Fréchet-Urysohn. 3. (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 = κ where κ is any cardinal ⩾ ℵ 2 satisfying κ = 2 < κ , and there is a hereditarily normal γ N -space.