Recently, Levin and Saxon , De Wilde and Houet  defined the σ-barrelledness while Husain  defined the countable barrelledness and countable quasibarrelledness. It is well-known that barrelled spaces are countably barrelled, and countably barrelled spaces are σ-barrelled. It is natural to ask whether there is some condition for σ-barrelled (resp. countably barrelled) spaces to be countably barrelled (resp. barrelled). Using the concept of
S-absorbent sequences of sets, we are able to give such conditions in Theorem 2.5 and Corollaries 2.6 and 2.7.