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Does there exists a "simple" proof that every smooth map $f: M\to N$ between manifolds contains at least one regular value, without using Sard's theorem?

Motivation: In Milnor, Brouwer's fixed point theorem is proved by means of Sard's theorem, but it is only used for the claim that a map from an $(n+1)$-disc into the boundary sphere contains at least one regular point (if I understand the text correctly). This seems much weaker than Sard's theorem that states that almost all values are regular.

Peter Franek
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  • You want to assume $M$ is second countable (this may or may not be part of your definition of "manifold"). Otherwise you could take $M$ to be the disjoint union of continuum-many intervals, with a different constant value on each interval. – Robert Israel Nov 30 '16 at 20:39
  • @RobertIsrael Yes, usually it is part of the definitions.. I would be happy with some elementary argument just for the disc and sphere, however. – Peter Franek Nov 30 '16 at 20:40
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    I'd be surprised if something like this existed. Sard's theorem is an application of BCT directly and hence requires some essential use of Zorn's lemma (or choice). Any proof of existence which is somehow constructive avoiding this essential use of choice would be great. I don't know if people have thought about models of ZF where Sard is false, but I wouldn't be surprised if these existed. – PVAL-inactive Nov 30 '16 at 21:35

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