Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-58536Speical Issue on the Occasion of Banaschewski's 90th Birthday (I)20170101Choice principles and lift lemmas12114640448ENMarcelErn'eFaculty for Mathematics and Physics, IAZD, Leibniz Universit\"at, Welfengarten 1, D 30167 Hannover, Germany.Journal Article20160929We show that in ${bf ZF}$ set theory without choice, the Ultrafilter Principle (${bf UP}$) is equivalent to several compactness theorems for Alexandroff discrete spaces and to Rudin's Lemma, a basic tool in topology and the theory of quasicontinuous domains. Important consequences of Rudin's Lemma are various lift lemmas, saying that certain properties of posets are inherited by the free unital semilattices over them. Some of these principles follow not only from ${bf UP}$ but also from ${bf DC}$, the Principle of Dependent Choices. On the other hand, they imply the Axiom of Choice for countable families of finite sets,which is not provable in ${bf ZF}$ set theory.http://cgasa.sbu.ac.ir/article_40448_f354e76a770fa82f66fa30955e1aba56.pdf