Reinventing Weak Barrelledness

With scores of novel theorems / examples we establish a hierarchy of 16 barrelled-type properties, consolidating decades of work by dozens of authors, defining / motivating / displaying optimal results. We prove our display answers all 4.29 billion interrelational questions at a glance. We solve the countable enlargement problems for separable weak barrelledness, uniformly relax certain Valdivia hypotheses to a characterization, and show that the existence of measurable cardinals depends on which of two hypotheses is optimal for Dierolf's dense subspace theorem.