16
$\begingroup$

It's widely known$^{1}$ that in the proof of Urysohn's Lemma (UL) one uses the Principle of Dependent Choice (DC). Inspired by the equivalence between DC and Baire's Category Theorem$^{2}$, I'd like to know if Urysohn's Lemma implies Dependent Choice, i.e.,

$ZF + DC \leftrightarrow ZF + UL$.

References:

(1.) Thomas Jech, The Axiom of Choice, Dover Publications, 2008, ISBN-13: 978-0486466248.

(2.) Charles Blair, The Baire category theorem implies the principle of dependent choices, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., v. 25 n. 10 (1977), pp. 933–934.

Improve this question
$\endgroup$
4
  • 3
    $\begingroup$ According to Andreas Blass’ comment in mathoverflow.net/questions/95257, Urysohn’s lemma is provable using dependent multiple choice, which I take to mean that it is weaker than DC. $\endgroup$
    – Emil Jeřábek
    Jun 12, 2012 at 16:55
  • $\begingroup$ Reference: jstor.org/stable/1998165 . $\endgroup$
    – Emil Jeřábek
    Jun 12, 2012 at 17:06
  • 3
    $\begingroup$ By the way, the article by Charles Blair (Baire for complete metric spaces implies DC) is remarkable (perhaps even unique?) in the following sense: The proof is so short that J.Rubin could give the whole proof in her review MR0469765 (57 #9546) in Math Reviews. $\endgroup$
    – Goldstern
    Jun 12, 2012 at 22:17
  • 3
    $\begingroup$ As far as I know it is an open problem whether DMC implies DC in ZF, as pointed out, for instance at Fossy-Morillon: "The Baire category property and some notions of compactness" (by the end of the first page) and also at the "Consequences of the axiom of choice" webpage. Hence, also the OP's question seems to be open as well. It is known however, that both Urysohn's lemma and DMC are strictly weaker than DC in ZF theory with atoms, since for example, in Levy's first permutation model N6, the first two hold while the last fails. $\endgroup$
    – godelian
    Jun 12, 2012 at 22:29

1 Answer 1

Reset to default
12
$\begingroup$

In Versions of normality and some weak forms of the axiom of choice Paul Howard et al exhibit a model of MC (Multiple Choice) and not-DC, see page 381. In that model Urysohn's Lemma (NU) holds, so it does not imply the Principle of Dependent Choices.

Improve this answer
$\endgroup$
1
  • $\begingroup$ The model you are referring to is a permutation model. It means that in ZFA, UL is strictly weaker than DC, but the question on whether the same holds in ZF seems to be still open (as it is still open whether DMC implies DC in ZF). $\endgroup$
    – Lorenzo
    Feb 13, 2023 at 11:26

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.