Exercises relate to the following definition: Let (X, d) be a metric space. A subset D of X is said to be dense in X if cl D=X.
Prove that ℚ × ℚ = {(x, y) : x ∈ ℚ and y ∈ ℚ } is dense in ℝ2 with the Euclidean metric. (Since ℚ × ℚ is countable, this means that ℝ2 is separable. See Exercise)
Exercise:
Prove that ℚ is dense in ℝ with the usual absolute value metric. [If a metric space (X, d) has a countable subset that is dense, then X is said to be separable. Thus in this exercise you are to show that ℝ is separable.]
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