On spaces with k-in-countable bases or weak bases

We prove that, for any k∈N, every regular star compact space with a k-in-countable base is metrizable. We also provide a metrization theorem for compact spaces with 2-in-finite weak bases; this gives a partial answer to a question of Bennett and Martin. It turns out that, for any m∈N, if X has an m-in-countable base and weak countable tightness (or k-property), then X is strongly monotonically monolithic. We apply this result to show that an example of Davis, Reed and Wage provides a consistent negative answer to a problem of Tkachuk.