What do you exactly mean by 2 examples ?

Well, one example could be R. It's clearly complete and it has disjoint dense subsets, namely Q and Qc (the complement of Q).

Do you want another example besides that one ?

Well, one example could be R. It's clearly complete and it has disjoint dense subsets, namely Q and Qc (the complement of Q).

Do you want another example besides that one ?

Example 2: R^2 with Q^2 and (Q^2)c. I can give you n more examples if you want, where n is a natural number.

What about the algebraic and tracsendent real numbers in R?

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