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5. (a) Is the pseudometric p in Exercise 3 equivalent to the usual pseudometric for the plane? Explain.

5. (a) Is the pseudometric p in Exercise 3 equivalent to the usual pseudometric for the plane? Explain.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Basic Pseudometric Spaces Q5:
3. Let X be the plane and for x = (₁, 2) and y = (y₁.92), let p(x, y) = ₁ - V₁ + 12-92|- (a) Show that p is a pseudometric. (b) Describe the p-cell of radius r centered at the point (a, b). Find the p-closure of S = {re X: r + x² < 1}. (e) 4. Let X be the plane. For r = (₁, 2) and y = (v₁.32) in X, let o(z,y) = |1 - ₁. (a) Show that is a pseudometric. (b) Describe the o-cell of radius r centered at the point (a, b). (e) Find the o-closure of S = {re X: +1} < 1}. 5. (a) Is the pseudometric p in Exercise 3 equivalent to the usual pseudometric for the plane? Explain.
Transcribed Image Text:3. Let X be the plane and for x = (₁, 2) and y = (y₁.92), let p(x, y) = ₁ - V₁ + 12-92|- (a) Show that p is a pseudometric. (b) Describe the p-cell of radius r centered at the point (a, b). Find the p-closure of S = {re X: r + x² < 1}. (e) 4. Let X be the plane. For r = (₁, 2) and y = (v₁.32) in X, let o(z,y) = |1 - ₁. (a) Show that is a pseudometric. (b) Describe the o-cell of radius r centered at the point (a, b). (e) Find the o-closure of S = {re X: +1} < 1}. 5. (a) Is the pseudometric p in Exercise 3 equivalent to the usual pseudometric for the plane? Explain.
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Step 1

5) 

No, the pseudometric defined by d(x,y) = 2|x1-y1|+|x2-y2| is not equivalent to the usual pseudometric on the plane.

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