Base multiplicity in compact and generalized compact spaces

We show that a compact Hausdorff space is metrizable if it has a base B such that every countably infinite subset of X is contained in at most countably many members of B. We show that the same statement for countably compact spaces is consistent with and independent of ZFC. These results answer questions stated by Arhangelʹskii et al. [Topology Appl. 100 (2000) 39–46]. We prove some strenthenings of these theorems. We also consider generalizations of our results to higher cardinalities as well as to wider classes of spaces.