Geodesic
Choice principles in elementary topology and analysis
Commentationes Mathematicae Universitatis Carolinae, Tome 38 (1997) no. 3, pp. 545-552.

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Many fundamental mathematical results fail in {\bf{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 {\it precisely} to validate each of certain basic analytical and topological results.
Classification : 03E25, 04A25, 26A03, 26A15, 54A35, 54B10, 54C35, 54D20, 54D30, 54D65, 54E45, 54E50, 54E52
Keywords: Axiom of (Countable) Choice; Boolean Prime Ideal Theorem; Theorems of Ascoli; Baire; Čech-Stone and Tychonoff; compact; Lindelöf and orderable spaces 
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Herrlich, Horst. Choice principles in elementary topology and analysis. Commentationes Mathematicae Universitatis Carolinae, Tome 38 (1997) no. 3, pp. 545-552. http://geodesic.mathdoc.fr/item/CMUC_1997__38_3_a9/