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Continuous functions can be distinguished by their values at say rational points of [0 1]. Germs of analytic functions can be distinguished by derivatives at a point. So in both cases we see countable sets of "parameters" for out set.

Consider set of germs of continuous functions.

Question: Is there a countable set of parameters such that different germs have different values of params ?

If yes, is there any "nice" set of parameters ? Or one may prove existence, but impossible to construct them "constructively / explicitly" ?

Related question: "Nice" functions on germs of continuous functions. My informal question is: "How to thing of germs of continuous functions/ what are "coordinates" on that space ? ". Any informal comments are welcome.

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The way I view it, germs of continuous functions are like

  • tails of infinite binary sequences, or
  • real numbers up to rational translation.

Finding "coordinates" on the space of germs would be like finding an explicit Vitali set.

Possible using the Axiom of Choice, but not in any explicit way.

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  • $\begingroup$ Thank you ! First sentence : analytic - you mean continuous? $\endgroup$
    – Alexander Chervov
    Commented Oct 22, 2014 at 19:19
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    $\begingroup$ Thanks for adding the relevant tags Bjørn! Right now, I only have time for my daily sweep of MO but the relevant keywords are $E_0$ and Borel equivalence relations. $\endgroup$
    – François G. Dorais
    Commented Oct 23, 2014 at 0:03
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    $\begingroup$ Yes, and because $E_0$ is a minimal degree above smooth, it suffices to show that this equivalence is not smooth. $\endgroup$
    – François G. Dorais
    Commented Oct 23, 2014 at 2:59
  • $\begingroup$ It seems you are saying that germs are isomorphic as a ring to tails of sequences with additional requirement that sequences should have a limit. Indeed, take x_n->0 and map a function to its values f(x_n), passing to germs corresponds to tails. Correct ? What is the role of AC ? $\endgroup$
    – Alexander Chervov
    Commented Oct 25, 2014 at 18:17
  • $\begingroup$ @AlexanderChervov we can look at the values $f(x)$ for $x\in\mathbb Q$, $x$ close to $0$, to determine the germ of $f$ at 0. But to pick out one and only one representative of each germ requires choice $\endgroup$
    – Bjørn Kjos-Hanssen
    Commented Oct 25, 2014 at 18:53

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