Countably enlarging weak barrelledness

Abstract

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 $\aleph₀$-dimensional subspace $M⊂ E^*$ 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 $CE$s 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 $\aleph₀$- barrelled and dual locally complete, inclusive, under the $ZFC$-consistent assumption that $\aleph₁ < b$.