Arhangel'skiĭ's solution to Alexandroff's problem: A survey

In 1969, Arhangel'skiĭ proved that |X|⩽2χ(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 Arhangel'skiĭ's 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.