4.3 Give an example to show that the arbitrary intersection of open sets is not anopen set.4.4 Prove that if A is nowhere dense in discrete space (S, d) then A = Ø.4.5 Prove that a closed set in the metric space (S, d) either is nowhere dense in S or elsecontains some nonempty open set. 2.2 Given (R", d) with d : R" × R" → R defined bynd(х, у) —(Lk – Yk)².=-k=1Determine whether d satisfies the triangle inequality or not. [ x = (x1, F2, . .. , In) andy = (y1, 92, ... , Yn) are elements of R"-]2.3 Let (S, d) be a metric space and suppose that p: S x S → R is defined byd(x, y)1+ d(x, y)p(x, y)for all points x, y E S.2.3.1 Determine whether p satisfies the triangle inequality or not.2.3.2 Show that (S, p) is bounded and that p(x, y) < d(x, y) for all x, y E S.

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Advanced Math

4.3 Give an example to show that the arbitrary intersection of open sets is not an open set.

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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Concept explainers

Set Theory

Every field in the present world involves a set. The theory of sets was developed by German mathematician Georg Cantor while working on the “problems of trigonometric series”.

Question

4.3 Give an example to show that the arbitrary intersection of open sets is not an
open set.
4.4 Prove that if A is nowhere dense in discrete space (S, d) then A = Ø.
4.5 Prove that a closed set in the metric space (S, d) either is nowhere dense in S or else
contains some nonempty open set.

2.2 Given (R", d) with d : R" × R" → R defined by
n
d(х, у) —
(Lk – Yk)².
=
-
k=1
Determine whether d satisfies the triangle inequality or not. [ x = (x1, F2, . .. , In) and
y = (y1, 92, ... , Yn) are elements of R"-]
2.3 Let (S, d) be a metric space and suppose that p: S x S → R is defined by
d(x, y)
1+ d(x, y)
p(x, y)
for all points x, y E S.
2.3.1 Determine whether p satisfies the triangle inequality or not.
2.3.2 Show that (S, p) is bounded and that p(x, y) < d(x, y) for all x, y E S.

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