Metrizable Barrelled Countable Enlargements

In 1980, Robertson, Tweddle and Yeomans solved the metrizableBCE problem in certain special cases, for example, in the normablecase under assumption of the Continuum Hypothesis (CH). A completepositive solution was recently given by the author and L. M.Sánchez Ruiz under an assumption, weaker than CH, thatscales exist. A better partial solution for the general BCEproblem obviates scales or other extra-ZFC assumptions, andentirely lays to rest the metrizable BCE problem. The proofstimulates thought about how the least infinite-dimensionalityof normed barrelled spaces impinges on the separable quotientproblem for Banach spaces. 1991 Mathematics Subject Classification46A08.