Let X = ℝ2, let d be the Euclidean metric, let d1 be the metric of Example 3.6.5, and let...

Let X = ℝ2, let d be the Euclidean metric, let d1 be the metric of Example 3.6.5, and let d2 be the metric of Exercise 3. Let A be a subset of ℝ2. Prove that A is open in (ℝ2,d) iff A is open in (ℝ2, d1) iff A is open in (ℝ2,d2). [Two metrics for a set are said to be (topologically) equivalent if a subset is open with respect to one metric iff it is open with respect to the other. Thus in this exercise you are to show that d, d1, and d2 are equivalent metrics.]

Exercise:

Let X = ℝ2 and define d2: ℝ2 × ℝ2 →ℝ by

d2((x1, y1), (x2, y2)) = max{|x1 –x2|, |y1 − y2|}

(a) Verify that d2 is a metric on ℝ2.

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(b) Draw the neighborhood N(0; 1) for d2, where 0 is the origin in ℝ2.