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compact space; first countable space; character of a point
We define a compactum $X$ to be AB-compact if the {\it 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. We strengthen several results from [Arhangel'skii and Buzyakova, {\it Convergence in compacta and linear Lindelöfness\/}, Comment. Math. Univ. Carolin. {\bf 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\`echet-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
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