We study the topological properties of minimally generated algebras (as introduced by Koppelberg) and, particularly, the subclass of T-algebras (a notion due to Koszmider) and its connection with Efimov's problem.We show that the class of T-algebras is a proper subclass of theclass of minimally generated Boolean algebras. It is also shown thatbeing the Stone space of a T-algebra is not even finitelyproductive.We prove that the existence of an Efimov T-algebra implies theexistence of a counterexample for the Stone-Scarborough problem. Wealso show that the Stone space of an Efimov T-algebra does not maponto the product (ω1+1)×(ω+1). We establish the following consistency results. Under CH there existsan Efimov minimally generated Boolean algebra; there are EfimovT-algebras in the forcing extensions obtained by adding ω2Cohen reals to any model of CH.
