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.,

$ZF + DC \leftrightarrow ZF + UL$.


(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.

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    $\begingroup$ 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 '12 at 16:55
  • $\begingroup$ Reference: jstor.org/stable/1998165 . $\endgroup$ – Emil Jeřábek Jun 12 '12 at 17:06
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    $\begingroup$ 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 '12 at 22:17
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    $\begingroup$ 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 '12 at 22:29

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