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Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.
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Is it meaningful to work on convergencies, integration, etc. on the Zariski topology?
I've decided to expand my comment into an answer.
The point is what do you think a topology is for. If you think that a topology is for talking about convergence of sequences, then no the Zariski top …