Some remarks on countably barrelled and countably quasibarrelled spaces

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Barrelled and quasibarrelled spaces form important classes of locally convex spaces. In (2), Husain considered a number of less restrictive notions, including infinitely barrelled spaces (these are the same as barrelled spaces), countably barrelled spaces and countably quasibarrelled spaces. A separated locally convex space E with dual E' is called countably barrelled (countably quasibarrelled) if every weakly bounded (strongly bounded) subset of E' which is the countable union of equicontinuous subsets of E' is itself equicontinuous. It is trivially true that every barrelled (quasibarrelled) space is countably barrelled (countably quasibarrelled) and a countably barrelled space is countably quasibarrelled. In this note we give examples which show that (i) a countably barrelled space need not be barrelled (or even quasibarrelled) and (ii) a countably quasibarrelled space need not be countably barrelled. A third example (iii)shows that the property of being countably barrelled (countably quasibarrelled) does not pass to closed linear subspaces.