Hereditary normality versus countable tightness in countably compact spaces

AbstractA space X is said to be countably tight if, for each A ⊂ X and each point x in the closure of A, there is a countable subset B of A such that x is in the closure of B. We show that the statement, “every separable, compact, hereditarily normal space is countably tight” is independent of the usual axioms of set theory, and show that it is equivalent to “no version of γN is hereditarily normal”, where γN is a familiar type of space due to Franklin and Rajagopalan, and a number of other statements. We derive some consequences from the fact that PFA implies this statement, including the consistency of “every countably compact, hereditarily normal space is sequentially compact”