Abstract
Rudin's Lemma, which appeared more than forty years ago, is a consequence of the Axiom of Choice and has been playing fundamental roles in the study of quasicontinuity in domain theory. This note aims to revealing the fact that Rudin's Lemma (and its variants) and the Boolean prime ideal theorem are equivalent in Zermelo-Fraenkel Set Theory, which is choiceless Set Theory. Our main techniques rely on Erné's Separation Lemma for Locales and a variant of Rado's Selection Principle due to Cowen.
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