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Let $X:=\{f: \mathbb{C}\to \mathbb{C}\}$ be a class of total functions on $\mathbb{C}$ closed under composition, addition, multiplication, and scalar multiplication. Does there exist a topology on $\mathbb{C}$ making these functions and only these functions continuous?

If it's not true in general (it probably isn't), are there any interesting known cases where it is true?

Note: I emphasize total functions because we want them to be everywhere defined. This avoids functions with bad singularities.

Edit: Obviously, continuous functions in the standard topology fit this bill, but this is tautological and not in the spirit of the problem.

Edit 2: Apparently the way I asked this question made it seem like I was looking for an answer to the "general case" which seems pretty untrue although I haven't actually worked it out. Rather, the real question was interesting cases where it is true.

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closed as no longer relevant by Harry Gindi, Joel David Hamkins, Bjorn Poonen, Pete L. Clark, Tom Leinster Feb 13 '10 at 12:19

This question is unlikely to help any future visitors; it is only relevant to a small geographic area, a specific moment in time, or an extraordinarily narrow situation that is not generally applicable to the worldwide audience of the internet. For help making this question more broadly applicable, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

Example machine: Take X to be the set of all continuous function for a topology on C with respect to which addition and multiplication are continuous... – François G. Dorais Feb 13 '10 at 6:11
OK, here are 2^{2^{\aleph_0}} examples: Let s be any field automorphism of C, and consider {s \circ f \circ s^{-1} : f is continuous in the usual sense}! – Bjorn Poonen Feb 13 '10 at 6:15
Could you guys close this? I know at least Joel and Pete can vote to close. – Harry Gindi Feb 13 '10 at 6:15
I guess I didn't put enough thought into the question, as Pete revealed to me via the Socratic method. – Harry Gindi Feb 13 '10 at 6:21
@JDH: We are all finding our way in this new social environment, so what I say is tentative, but: it seems to me that if the questioner wants a question to be closed, it should be -- and stay -- closed. I think Harry is right when he says that anyone who is interested in the question can easily ask a new (and, one hopes, improved) version of the question. Please go ahead and ask a new question if you like: the topic seems interesting to me too. – Pete L. Clark Feb 14 '10 at 1:57

1 Answer 1

If X is the set of all functions, then it has your closure properties, and the indiscrete topology {emptyset, C} makes exactly those functions continuous.

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