Rudin's Lemma and Reverse Mathematics

Abstract

Domain theory formalizes the intuitive ideas of approximation and convergence in a very general way, and provides a fundamental tool in the study of computing theory and computability theory. In search of complete lattices on which the Laswon topology is Hausdorff, Gierz, Lawson and Stralka introduced in [3] quasicontinuous lattices, which inherit many good properties of domains. Gierz, et al. pointed in [3] that Rudin's Lemma for finding a “cross-section” of certain descending family of sets plays a central role in the development of the whole theory of quasicontinuous lattices. In this paper, we study Rudin's Lemma from reverse mathematics point of view and prove that the Rudin's Lemma is equivalent to ACA₀ over RCA₀.