Hereditary normality versus countable tightness in countably compact spaces (Q1203843)

This paper investigates the question of the existence of separable countably compact hereditarily normal spaces of uncountable tightness. The author notes that no such space exists if \(2^{\aleph_ 0}<2^{\aleph_ 1}\). There are essentially two main results. The first is that the existence of such a space is equivalent to the existence of a hereditarily normal ``Franklin-Rajagopalan space'' (i.e. a locally compact extension, \(\gamma N\), of the countable discrete space \(N\) whose remainder is homeomorphic to the ordinal space \(\omega_ 1\)). The one- point compactification of a hereditarily normal \(\gamma N\) will be a compact hereditarily normal space of uncountable tightness. The second result is a clever argument to establish the consistency of the existence of such spaces. The model constructed is a model of Martin's Axiom (MA) plus \(2^{\aleph_ 0}=\aleph_ 3\) (and requires that the cub filter on \(\omega_ 1\) is \(\omega_ 2\)-generated). It is remarked that it has been shown to follow from the Open Coloring Axiom (OCA) that there is no hereditarily normal \(\gamma N\) (hence this is consistent with MA plus \(2^{\aleph_ 0} = 2^{\aleph_ 1} = \aleph_ 2\)) and that the existence of a hereditarily normal \(\gamma N\) is consistent with MA plus \(2^{\aleph_ 0} = \aleph_ 2\).