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Let (X, d) be a metric space. Prove each of the following: (i) A closed ball is closed. (ii) Intersection of two open sets is open (iii) If p(x, y) = 1+d(x,y)' d(x,y) then p(x, y) is a metric on X.

Let (X, d) be a metric space. Prove each of the following: (i) A closed ball is closed. (ii) Intersection of two open sets is open (iii) If p(x, y) = 1+d(x,y)' d(x,y) then p(x, y) is a metric on X.

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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3. Let (X, d) be a metric space. Prove each of the following: (i) A closed ball is closed. (ii) Intersection of two open sets is open (iii) If p(x, y) = = d(x,y) 1+d(x,y) ¹ then p(x, y) is a metric on X. 4. Let XRx R, where R is the set of reals and let d be the Euclidean metric on X. Show that the set {(x, y) = Xx² + y² < 1} is open.
Transcribed Image Text:3. Let (X, d) be a metric space. Prove each of the following: (i) A closed ball is closed. (ii) Intersection of two open sets is open (iii) If p(x, y) = = d(x,y) 1+d(x,y) ¹ then p(x, y) is a metric on X. 4. Let XRx R, where R is the set of reals and let d be the Euclidean metric on X. Show that the set {(x, y) = Xx² + y² < 1} is open.
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