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.
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.