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I am currently working on the axiom of choice and was looking for easy applications. A common example is the proof of the equivalence of continuity with continuity in terms of limits of sequences. Usually the proof uses the axiom of choice.

I found this site (http://www.apronus.com/math/cauchyheine.htm), where it is shown that one does not need the axiom of choice, if f is continuous in a neighbourhood of x. It is claimed though, that we do need the axiom of choice in the general case.

Now I was wondering how one can show that the axiom of choice is necessary for this proof, and whether the proposition is equivalent to some version of the axiom of choice, like countable choice, or the statement that there is a choice function for the real numbers?

thank you for your replies

boukkoun
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  • That was asked before on the site. – Asaf Karagila Jul 07 '13 at 12:55
  • http://math.stackexchange.com/questions/126010/continuity-and-the-axiom-of-choice also see http://math.stackexchange.com/questions/346526/proof-of-a-basic-ac-omega-equivalence for showing how to prove countable choice from this sort of assumption on metric spaces. – Asaf Karagila Jul 07 '13 at 12:56
  • I am sorry I didn't find that post. Thank you very much for your answer! – boukkoun Jul 07 '13 at 13:14

1 Answers1

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There are two ways to see that the axiom of choice is needed for proving a certain statement:

  1. Construct a model in which the axiom of choice fails, and the statement fails.
  2. Assume that the statement is true, and prove a weak form of choice.

In the case of continuity of real valued functions it is not hard to construct a counterexample using standard models of $\lnot\sf AC$. It is slightly (but not much) harder to prove that every $f\colon\Bbb{R\to R}$ is sequentially continuous at $x$ if and only if it is continuous at $x$, implies the axiom of choice for countable sets of real numbers.

The general proof that a function between metric spaces is continuous if and only if it is sequentially continuous implies the axiom of countable choice, is not too difficult either (at least to read, I'm sure that coming up with it was difficult).

For more details you can, and should, check out Herrlich's wonderful book The Axiom of Choice where he covers these questions in details.

Asaf Karagila
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