Let X = ℝ2 and let d be the Euclidean metric. Define d3: ℝ2 × ℝ2 →ℝ by d3(x, y) = min {1,...

Let X = ℝ2 and let d be the Euclidean metric. Define d3: ℝ2 × ℝ2 →ℝ by d3(x, y) = min {1, d(x, y)}. (This metric is sometimes called the “nearsighted” metric. Up to one unit away, every point is distinguished clearly. But past one, everything gets blurred together.)

(a) Verify that is a metric on ℝ2.

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(b) Draw the neighborhoods , N(0, 1), and N(0, 2) for d3, where 0 is the origin in ℝ2.