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

Advanced Engineering Mathematics
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Chapter2: Second-order Linear Odes
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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.
Transcribed Image Text: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.
Transcribed Image Text: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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