MATH 521, Spring 2014, Assignment 3Due date: Wednesday, February 26 Monday, March 3Problems for submission:1. Prove that the following are metric spaces:(a) (Rn, d) where, for all x, y ∈ Rn,d(x, y) =0, if x = y1, otherwise.(b) (Rn, d∞) where d∞(x, y) = maxi=1,...,n|xi− yi| for x, y ∈ Rn.(c) (Rn,˜d) where˜d(x, y) =d(x, y)1 + d(x, y)for x, y ∈ Rnwhere d(x, y) is any metric on Rn. (Hint: Provefirst of all that b ≥ a ≥ 0 implies thatb1 + b≥a1 + a!)2. Consider the three metrics in Question #1 with R2. Draw or describethe unit ball B1(0) around the point 0 = (0, 0).3. Consider the Hausdorff metricd(I1, I2) = maxmaxx∈I1miny∈I2|x − y|, maxy∈I2minx∈I1|x − y|on the space of intervals I = {[a, b] | a, b ∈ R}.(a) Determine the distance between the following pairs of sets:(i) I1= [0, 1] and I2= [1, 2], (ii) I1= [0, 1] and I2= [−2, 2].(b) Prove that d(I1, I2) satisfies the first two metric properties. (Youdo not need to prove the ∆-inequality!)4. Consider the metric space (R, d) with the metric d(x, y) = |x − y|. Forthe following subsets S ⊆ R, identify the sets So, S0, ∂S, and S. Alsodetermine whether the sets S are open, closed, both, or neither.(a) S = {x ∈ R | |x − y| < r} where r > 0 and y ∈ R is fixed.(b) S =1n+1m∈ R | n, m ∈ N.(c) S = {x ∈ R | 0 < x < 1, x is irrational}.5. Suppose (X, d) is a metric space and S ⊆ X. Prove the following:(a) S0is closed.(b) S = S ∪ ∂S.(c) Sc= (So)c(d) ∂S = S ∩ Sc6. Suppose (X, d) is a metric space and A, B ⊆ X. Prove the following:(a) (A ∩ B)o= Ao∩ Bo(b) (A ∪ B)o⊇ Ao∪ Bo(c)A ∩ B ⊆ A ∩ B(d) A ∪ B = A ∪ B(e) Find sets A, B ⊆ R such that the inclusions (b) and (c) are strict.Honors Question #1 Consider a metric space (X, d) equipped with the discrete metric d(x, y) =0 for x = y and d(x, y) = 1 for x 6= y.(a) What are the open subsets S ⊆ X in this topology? What arethe closed sets? Justify your claims.(b) Is X connected? Why or why not.Honors Question #2 Is it true that So= (S)o? Either prove the claim that it is true or finda