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For the definition of moduli space of Riemann surface, it seems that it is not based on the definiton in terms of functor(representable...) but rather put a topology on the set of isormophism classes. Is there any difference between these? Or if I miss something? Is there any reference ? Thank you!

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    Possibly this might help? http://en.wikipedia.org/wiki/Equidistribution_theorem2010-09-30

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