AB - compacta Isaac

We define a compactum X to be AB-compact if the cofinality of the character χ(x, Y ) is countable whenever x ∈ Y and Y ⊂ X. It is a natural open question if every AB-compactum is necessarily first countable. We strengthen several results from [Arhangel’skii and Buzyakova, Convergence in compacta and linear Lindelöfness, Comment. Math. Univ. Carolin. 39 (1998), no. 1, 159–166] by proving the following results. (1) Every AB-compactum is countably tight. (2) If p = c then every AB-compactum is Frèchet-Urysohn. (3) If c < אω then every AB-compactum is first countable. (4) The cardinality of any AB-compactum is at most 2 .