Metric Space Exercises

You may work alone or in a group of up to 3 people and turn in 1 per group. Make copies of your work so that you are prepared to present any of the exercises after you turn it in.

Problem 1

Use the following from Sibley's Geometric Viewpoint to help you, but write out a complete proof that SAS does not hold in R2 with the taxicab metric.

Problem 2

Mendelson p. 34 # 3

Problem 3

Let (X,d) and (Y,p) be metric spaces. Assume that f: X --> Y is an isometry, ie that f preserves distances. Prove that f is continuous.

Problem 4

Let (X,d) be a metric space, x in X and Epsilon > 0. Prove that B_d (x, Epsilon) is an open set.

Graduate Problem

Mendelson p. 35 #8

Giving Proper References

Be sure to give proper reference where it is due. Even if you find ideas or a proof elsewhere, all work needs to be written up in your own (group's) words and explanations, although you should still reference the original work or person.