 # Problem 4. Prove care about the lengths of distances, even if poin...

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Problem 4. Prove care about the lengths of distances, even if points X in with the metric space become arbitrarily far away. More specifically, prove the that topology doesn't following. Given any metric space d: (a) the function d(x,y) = 1+d(x,y) d(x,y) is a metric on X. (b) the metrics à and d are equivalent (and therefore they generate the same topology on X). Problem 5. Consider a circular disk {(x, y) < 1} of unit radius centered at (0,1). We want to slide the vertical chords of this disk down so their lower endpoints are on the x-axis. (a) Show that the image of the disk will be the upper half of an elliptical region 2 + y 4 2 (b) In terms of the coordinates of a point (x,y) in the disk, how far must we slide the point? (c) Write a formula for the sliding function S : disk ellipse. (d) Show that S is a homeomorphism.

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