Countably enlarging weak barrelledness

Abstract

If is a locally convex space with dual and is the coarsest topology finer than such that the dual of is for a given -dimensional subspace transverse to , then is a countable enlargement of . Here most barrelled results are optimally extended within the fourteen properties introduced in the 1960s, '70s, '80s, '90s and recently studied in "Reinventing weak barrelledness", et al. If a exists, one exists with none of the fourteen properties. Yet s that preserve precise subsets of these properties essentially double the stock of distinguishing examples. If a exists, must one exist that preserves a given property enjoyed by ? Under metrizability, the fourteen cases become two: the metrizable question we answered earlier, and the metrizable inductive question we answer here (both positively). Without metrizability we are as yet unable to answer Robertson, Tweddle and Yeomans' original question (1979), the question and four others. We give negative answers for the eight remaining general cases, those between - barrelled and dual locally complete, inclusive, under the -consistent assumption that .