5. Let (X, d) be a metric space.i) Show that the open ball B, (xo) = {x € X : d (x, xo) < r} is an open set in (X, d).ii) Show that the closed ball C, (xo)={x € X : d (x, xo)

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Answered: 5. Let (X, d) be a metric space. i)…

5. Let (X, d) be a metric space. i) Show that the open ball B, (ro) = {x € X : d (T, xo) < r} is an open set in (X, d). ii) Show that the cloeod boll

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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5. Let (X, d) be a metric space.
i) Show that the open ball B, (xo) = {x € X : d (x, xo) < r} is an open set in (X, d).
ii) Show that the closed ball C, (xo)={x € X : d (x, xo) <r} is a closed set in (X, d).
iii) Show that cl (B, (xo)) C C, (xo). State any property that you use. Show that this
inclusion can be strict if d is the discrete metric.

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