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When doing analysis with infinitesimals, we can define continuous functions as those that preserve infinitesimal distances. But this only works for maps between manifolds - is there a good way to recover a full topology?
When doing analysis with infinitesimals, we can define continuous functions as those that preserve infinitesimal distances. But this only works for maps between manifolds - is there a good way to recover a full topology?
Eg, if we say a closed subset of X is {f(x) = 0} for some continuous (in the above sense) f: X --> R, is it true that functions are continuous iff preimages of closed sets are closed?