Chain-condition methods in topology

The special role of countability in topology has been recognized and commented upon very early in the development of the subject. For example, especially striking and insightful comments in this regard can be found already in some works of Weil and Tukey from the 1930s (see, e.g., Weil (1938) and Tukey (1940, p. 83)). In this paper we try to expose the chain condition method as a powerful tool in studying this role of countability in topology. We survey basic countability requirements starting from the weakest one which originated with the famous problem of Souslin (1920) and going towards the strongest ones, the separability and metrizability conditions. We have tried to expose the rather wide range of places where the method is relevant as well as some unifying features of the method.