Let (X, d) be a metric space. In this problem we will show that there is a bounded metric onX that does not change the open/closed/compact sets of (X, d). This is a more general wayto see that a closed and bounded set may not be compact (as every closed set is boundedwith the new metric).(a) Show that the function d, : X × X → R defined by d1(p, q)on X.d(p.q)_ is also a metricd(p,q)+1(b) Show that in (X, d1), X is a bounded metric space, regardless of whether (X, d) is abounded metric space.(c) Show that UC X is open in (X, d) if and only if U is open in (X, d1). Hint: Show thatthe function f(x) = is one to one.x+1

Let (X, d) be a metric space. In this problem we will show that there is a bounded metric on X that does not change the open/closed/compact sets of (X, d). This is a more general way to see that a closed and bounded set may not be compact (as every closed set is bounded vith the new metric). (a) Show that the function d1 : X × X → R defined by d1(p, q) = on X. d(p,q) d(p,q)+1 is also a metric

Let (X, d) be a metric space. In this problem we will show that there is a bounded metric on X that does not change the open/closed/compact sets of (X, d). This is a more general way to see that a closed and bounded set may not be compact (as every closed set is bounded vith the new metric). (a) Show that the function d1 : X × X → R defined by d1(p, q) = on X. d(p,q) d(p,q)+1 is also a metric

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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