Exercise 1.31: In â„â¿, let R denote the set of points having only rational coordinates and â„“ its complement, that is, the set of points having at least one irrational coordinate. Then prove that R° = ℓ° = ∅.
Let R be the same as in Exercise 1.31. Prove that every nonempty open set in â„â¿ contains a member of R. Also, prove that every nonempty open set in â„â¿ contains infinitely many members of R. Solve this question.