Let X = R2 and let d be the Euclidean metric. Define d3: R2 x R2 R by d3(x, y) min (1, d(x, y)). (This metric is sometimes called the "near- sighted" metric. Up to one unit away, every point is distinguished clearly. But past one, everything gets blurred together) A) Verify that d3 is a metric on R2.
B) Draw the neighborhoods N(0:), N(0:1), and N(0:2) for ds, where 0 is the origin in R2.