Efimov's problem and Boolean algebras (Q386145)

Associated with certain trees \(T\) is a \(T\)-algebra, which is a minimally generated Boolean algebra, introduced by Koszmider. The authors extend Koszmider's work on these algebras, relating them to Efimov's problem: Is there an infinite compact Hausdorff space which contains no infinite converging sequence and no copy of \(\beta\omega\)? An Efimov \(T\)-algebra is a \(T\)-algebra whose Stone space satisifies this condition. The main results are: (1) There is a minimally generated Boolean algebra which is not a \(T\)-algebra. (2) The existence of an Efimov \(T\)-algebra implies that there is a product of sequentially compact spaces which is not countably compact. (A counterexample to a conjecture of Scarborough and Stone.) (3) (Using Hechler forcing) The existence of an Efimov \(T\)-algebra of countable tightness is relatively consistent.