Weak barrelledness for C(X) spaces ✩

Buchwalter and Schmets reconciled Cc(X) and Cp(X) spaces with most of the weak barrelledness conditions of 1973, but could not determine if ℵ0-barrelled ⇔ ∞-barrelled for Cc(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 Cc(X) problem to its 1973 status and solve it by carefully translating the topology of Kunen (1980) and van Mill (1982) to find the example that eluded Buchwalter and Schmets. The more tractable Cp(X) readily partitions the conditions into just two equivalence classes, the same as for metrizable locally convex spaces, instead of the five required for Cc(X) spaces. Our paper elicits others, soon to appear, that analytically characterize when the Tychonov space X is pseudocompact, or Warner bounded, or when Cc(X) is a df -space (Jarchow’s 1981 question).  2004 Elsevier Inc. All rights reserved