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.