The Urysohn Lemma is independent of ZF + Countable Choice

Abstract

We prove that it is relatively consistent with Z F \mathsf {ZF} (i.e., Zermelo–Fraenkel set theory without the Axiom of Choice ( A C \mathsf {AC} )) that the Axiom of Countable Choice ( A C ℵ 0 \mathsf {AC}^{\aleph _{0}} ) is true, but the Urysohn Lemma ( U L \mathsf {UL} ), and hence the Tietze Extension Theorem ( T E T \mathsf {TET} ), is false. This settles the corresponding open problem in P. Howard and J. E. Rubin [Consequences of the Axiom of Choice, Mathematical Surveys and Monographs, Vol. 59, American Mathematical Society, Providence, RI, 1998]. We also prove that in Läuchli’s permutation model of Z F A \mathsf {ZFA} + + ¬ U L \neg \mathsf {UL} , A C ℵ 0 \mathsf {AC}^{\aleph _{0}} is false. This fills the gap in information in the above monograph of Howard and Rubin.