G. Show that a set A in R² is open in the Euclidean metric it is open in the Manhattan metric. Hint: As usual, there are two directions to prove in an . The picture on p73 of the notes may be somewhat helpful.

EC. Consider the closed rectangle R = [a,b] × [c,d] in ℝ2 (under the Euclidean metric).i. Explain why every horizontal line of the form [a,b] × {y} is compact.ii. Given any open cover Oλ of R, conclude that each line of the form [a,b] × {y} is covered by a finite subfamily of the Oλ's.iii. Using ii, show that R is compact.Hint: In iii, it may help to look at, for each horizontal line, the union of the finite subcover. How "thick" or "thin" can this union be?Note: Using the same argument as that for ℝ, it follows from this problem that a subset of ℝ2 is compact if and only if it is closed and bounded.

Please solve number 3