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.