Abstract:We define a compactum $X$ to be AB-compact if the {cofinality} of the character $\chi (x,Y)$ is countable whenever $x\in Y$ and $Y\subset X$. It is a natural open question if every AB-compactum is necessarily first countable. \par We strengthen several results from [Arhangel'skii and Buzyakova, {Convergence in compacta and linear Lindel\"ofness}, Comment. Math. Univ. Carolin. {39} (1998), no. 1, 159--166] by proving the following results. \roster \item Every AB-compactum is countably tight. \item If $\frak p = \frak c$ then every AB-compactum is Fr{\accent 18 e}chet-Urysohn. \item If $\frak c < \aleph _\omega $ then every AB-compactum is first countable. \item The cardinality of any AB-compactum is at most $2^{< \frak c}$. \endroster
Keywords: compact space, first countable space, character of a point
AMS Subject Classification: 54A20, 54A25, 54A35, 54D30