The purpose of this note is to give a rather brief survey of recent metrization theory while putting emphasis on developments during the last decade. R.E, Hodel's paper [23] is recommended for a quick survey of the early history of metrization theory as well as its development until 1972.
As is well known, a topological space is called metrizable iff it is homeomorphic to a metric space. Metrization theory is a field of topology where the main objective is to study conditions for a given topological space to be metrizable. In this note we assume that all spaces are at least T2 (Hausdorff), though some theorems may be true for more general spaces. Generally X denotes a T2-space. The reader is referred to [10], [40] for standard terminologies in general topology, while less popular concepts will be defined in the text.
CONDITIONS IN TERMS OF BASES. SEQUENCES OF COVERS AND SEQUENCES OF NEIGHBOURHOODS
In many classical theorems metrizability conditions were described in terms of bases (Urysohn's theorem, [40] p.196, Nagata-Smirnov theorem, p.194, Bing's theorem, p.198, Arhangelskii's theorem, p.203, etc.), sequences of open covers (Alexandroff-Urysohn theorem, p.184, Bing's theorem, p.199, etc.), sequences of nbds (= neighbourhoods) (Frink's theorem, p.192, Nagata's theorem, p.189, etc.) or similar entities. (Page numbers are, throughout, those of [40] where one can find the respective theorems.) The situation has changed little since then. As for metrizability conditions in terms of bases, the following result due to G. Gruenhage and P.J. Nyikos is especially interesting.