3. Prove each of the following for a metric space (X, d): (i) Union of two closed sets is closed. (ii) A set A CX is closed if and only if for each x E X-A, there is an open set U containing a such that UnA = 0. (iii) For A CX, x € clA, if and only if there is a sequence {n} A such that {n} →x.
3. Prove each of the following for a metric space (X, d): (i) Union of two closed sets is closed. (ii) A set A CX is closed if and only if for each x E X-A, there is an open set U containing a such that UnA = 0. (iii) For A CX, x € clA, if and only if there is a sequence {n} A such that {n} →x.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
3
![1. Give the statement of the Cantor's Intersection Theorem. Then give
examples to show that the assumptions in the hypothesis are necessary
to give the conclusion of the theorem.
2. Prove the Baire's Category theorem: "A complete metric space
(X, d) is a second category subset of itself."
Deduce from the above result in Question 2 that if {Un ne N} is
a sequence of open dense subsets of a complete metric space X, then
MEN Un # 0.
3. Prove each of the following for a metric space (X, d):
(i) Union of two closed sets is closed.
(ii) A set AC X is closed if and only if for each x EX-A, there is
an open set U containing a such that UnA = 0.
(iii) For A CX, xe clA, if and only if there is a sequence {n} A
such that {n} →x.
(iv) If a sequence {n} in X converges, it converges to a unique point.
(iv) Every convergent sequence in X is a Cauchy sequence.
(v) A subset Y of a complete metric space is (X, d) is complete if
and only if Y is closed and Y has the metric inherited from (X, d).
(vi) Show that the subset (0, 1] of the set of reals R with usual metric
is not complete.
4. Give the definition for a function f (X,d) → (Y, p) is continuous
at a point c E X using the distance functions d and p. Then translate
the definition in terms of open balls and also in terms of open sets.
Give an example of a function f (X, d) → (Y, p) which is continuous
and also a function which is not continuous., where neither the space
X nor Y is the set of Reals with usual metric.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faeb81122-6fbf-4c63-aaf5-71689ff4fea7%2F65ac8ff4-b93b-4b3d-94d0-e17e2fb241aa%2Fb8cqwhj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1. Give the statement of the Cantor's Intersection Theorem. Then give
examples to show that the assumptions in the hypothesis are necessary
to give the conclusion of the theorem.
2. Prove the Baire's Category theorem: "A complete metric space
(X, d) is a second category subset of itself."
Deduce from the above result in Question 2 that if {Un ne N} is
a sequence of open dense subsets of a complete metric space X, then
MEN Un # 0.
3. Prove each of the following for a metric space (X, d):
(i) Union of two closed sets is closed.
(ii) A set AC X is closed if and only if for each x EX-A, there is
an open set U containing a such that UnA = 0.
(iii) For A CX, xe clA, if and only if there is a sequence {n} A
such that {n} →x.
(iv) If a sequence {n} in X converges, it converges to a unique point.
(iv) Every convergent sequence in X is a Cauchy sequence.
(v) A subset Y of a complete metric space is (X, d) is complete if
and only if Y is closed and Y has the metric inherited from (X, d).
(vi) Show that the subset (0, 1] of the set of reals R with usual metric
is not complete.
4. Give the definition for a function f (X,d) → (Y, p) is continuous
at a point c E X using the distance functions d and p. Then translate
the definition in terms of open balls and also in terms of open sets.
Give an example of a function f (X, d) → (Y, p) which is continuous
and also a function which is not continuous., where neither the space
X nor Y is the set of Reals with usual metric.
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