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Buchwalter and Schmets reconciled C c (X) and C p (X) spaces with most of the weak barrelledness conditions of 1973, but could not determine if ℵ 0 -barrelled ⇔ ∞ -barrelled for C c (X). The areas grew apart. Full reconciliation with the fourteen conditions adopted by Saxon and Sánchez Ruiz needs their 1997 characterization of Ruess' property (L), which allows us to reduce the C c (X) problem to its 1973 status and solve it by carefully translating the topology of Kunen (1980) andvan Mill (1982) to find the example that eluded Buchwalter and Schmets. The more tractable C p (X) readily partitions the conditions into just two equivalence classes, the same as for metrizable locally convex spaces, instead of the five required for C c (X) spaces. Our paper elicits others, soon to appear, that analytically characterize when the Tychonov space X is pseudocompact, or Warner bounded, or when C c (X) is a df -space (Jarchow's 1981 question).

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While the separable quotient problem is famously open for Banach spaces, in the broader context of barrelled spaces we give negative solutions. Obversely, the study of pseudocompact X and Warner bounded X allows us to expand Rosenthal's positive solution for Banach spaces of the form C c (X) to barrelled spaces of the same form, and see that strong duals of arbitrary C c (X) spaces admit separable quotients.2010 Mathematics subject classification: primary 46A08; secondary 54C35.