The Urysohn Lemma is independent of ZF + Countable Choice

Abstract:

We prove that it is relatively consistent with $\mathsf {ZF}$ (i.e., Zermelo–Fraenkel set theory without the Axiom of Choice ($\mathsf {AC}$)) that the Axiom of Countable Choice ($\mathsf {AC}^{\aleph _{0}}$) is true, but the Urysohn Lemma ($\mathsf {UL}$), and hence the Tietze Extension Theorem ($\mathsf {TET}$), 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 $\mathsf {ZFA}$ $+$ $\neg \mathsf {UL}$, $\mathsf {AC}^{\aleph _{0}}$ is false. This fills the gap in information in the above monograph of Howard and Rubin.