Countably enlarging weak barrelledness


If (E, \xi) is a locally convex space with dual E' and η is the coarsest topology finer than \xi such that the dual of (E, η) is E'+M for a given \aleph<sub>0</sub>-dimensional subspace M⊂ E<sup>*</sup> transverse to E', then η is a countable enlargement (CE) of \xi. Here most barrelled CE(BCE) 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 CE exists, one exists with none of the fourteen properties. Yet CEs that preserve precise subsets of these properties essentially double the stock of distinguishing examples. If a CE exists, must one exist that preserves a given property enjoyed by \xi? Under metrizability, the fourteen cases become two: the metrizable BCE question we answered earlier, and the metrizable inductive CE(ICE) question we answer here (both positively). Without metrizability we are as yet unable to answer Robertson, Tweddle and Yeomans' original BCE question (1979), the ICE question and four others. We give negative answers for the eight remaining general cases, those between \aleph<sub>0</sub>- barrelled and dual locally complete, inclusive, under the ZFC-consistent assumption that \aleph<sub>1</sub> < b.

DOI Code: 10.1285/i15900932v17p217

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