Countably compact spaces with n-in-countable weak bases are metrizable

We prove that a Hausdorff compact space with an n-in-countable weak base is metrizable for each n ∈ N . This result gives a positive answer to a question of Bennett and Martin asking if a compact Hausdorff space with a 2-in-finite weak base is metrizable. We also discuss the properties of spaces with n-in-countable bases, monotonically monolithic spaces and spaces with property (G). Finally, we show that an example of Davis, Reed and Wage provides consistent negative answers to two problems raised by Tkachuk on monotonically monolithic spaces.