In 1969, Arhangel'skiȋ 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 Arhangel'skiȋ'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.
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