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
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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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
with 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 metric
d(p,q)+1
(b) Show that in (X, d1), X is a bounded metric space, regardless of whether (X, d) is a
bounded metric space.
(c) Show that UC X is open in (X, d) if and only if U is open in (X, d1). Hint: Show that
the function f(x) = is one to one.
x+1
Transcribed Image Text: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 with 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 metric d(p,q)+1 (b) Show that in (X, d1), X is a bounded metric space, regardless of whether (X, d) is a bounded metric space. (c) Show that UC X is open in (X, d) if and only if U is open in (X, d1). Hint: Show that the function f(x) = is one to one. x+1
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