Arhangelʹskiĭʹs solution to Alexandroffʹs problem: A survey

In 1969, Arhangelʹskiĭ proved that | X | ⩽ 2 χ ( X ) L ( X ) for every Hausdorff space X. This beautiful inequality solved a nearly fifty-year old question raised by Alexandroff and Urysohn. In this paper we survey a wide range of generalizations and variations of Arhangelʹskiĭʹs inequality. We also discuss open problems and an important legacy of the theorem: the emergence of the closure method as a fundamental unifying device in cardinal functions.