Abstract:Many fundamental mathematical results fail in {{ZF}}, i.e., in Zermelo-Fraenkel set theory without the Axiom of Choice. This article surveys results --- old and new --- that specify how much ``choice'' is needed {precisely} to validate each of certain basic analytical and topological results.
Keywords: Axiom of (Countable) Choice, Boolean Prime Ideal Theorem, Theorems of Ascoli, Baire, \v Cech-Stone and Tychonoff, compact, Lindel\"of and orderable spaces
AMS Subject Classification: 03E25, 04A25, 26A03, 26A15, 54-02, 54A35, 54B10, 54C35, 54D20, 54D30, 54D65, 54E45, 54E50, 54E52