Barrelled Countable Enlargements and the Dominating Cardinal

Extensively treated by Robertson, Tweddle, Yeomans, P´erez Carerras, and Bonet in the 1980s, the barrelled countable enlargement problem may be stated as follows: Let E be a barrelled locally convex space with E9/E*. Is there an /0-dimensional subspace M of E* transverse to E9 such that E endowed with the Mackey topology m E, E9qM.is also barrelled? While the general problem remains open, for E metrizable we give here a complete positive solution under the assumption that scales exist an assumption weaker than Martin’s Axiom..