Solve number 3 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) ¹then p(x, y) is a metric on X.4. Let XRx R, where R is the set of reals and let d be the Euclideanmetric on X. Show that the set {(x, y) = Xx² + y² < 1} is open.

Here we use some definitions like open set , closed set and metic space.

Answered: Let (X, d) be a metric space. Prove…

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

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Solve number 3

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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