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Does there exist a topological space X for which fine topology on C(X,R) with respect to usual metric on R and metric defined on R by the homeomorphism P: R .....> (-1,1) defined by P(x)=2/pi (arctanx) is respectively equal to Corresponding uniform topology on C(X,R) with respect to these metrics. But the uniform spaces C(X,R) with respect to these metrics are not homeomorphic?

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 I am sorry, but I really do not understand your question. You should probably rephrase it in several sentences. – Benoît Kloeckner Sep 20 2011 at 18:40
I don't think this is really a research question, so math.stackexchange might be a better place (see the FAQ). It would help if it was clearer though. – George Lowther Sep 20 2011 at 22:30
...I mean research level or of interest to professional mathematicians, which is the kind of question MO is intended for. – George Lowther Sep 20 2011 at 22:32
I agree with the previous comments, in that the question needs to be phrased more clearly (whether or not it is asked here, or on math.stackexchange) – Yemon Choi Sep 20 2011 at 23:59
If one can find such a example we can negate a certain conjecture. And regarding fine topology and uniform topology one can refer to Munkres book on general topology. – unknown (yahoo) Sep 24 2011 at 5:21

closed as too localized by Matthew Daws, Andres Caicedo, George Lowther, Bill Johnson, Gjergji Zaimi Sep 21 2011 at 19:05

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