1. Prove each of the following for a metric space (X, d): (i) A set A C X is closed if and only if for each x EX-A, there is an open set U containing x such that UnA = 0. (ii) For A CX, xa E clA, if and only if there is a sequence {n} CA such that {n} →x. (iii) 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 X is complete if and only if Y is complete and Y has the metric inherited from (X, d).
1. Prove each of the following for a metric space (X, d): (i) A set A C X is closed if and only if for each x EX-A, there is an open set U containing x such that UnA = 0. (ii) For A CX, xa E clA, if and only if there is a sequence {n} CA such that {n} →x. (iii) 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 X is complete if and only if Y is complete and Y has the metric inherited from (X, d).
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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Transcribed Image Text:1. Prove each of the following for a metric space (X, d):
(i) A set ACX is closed if and only if for each x E X-A, there is
an open set U containing x such that UnA = 0.
(ii) For A CX, x E clA, if and only if there is a sequence {n} A
such that {n} →x.
(iii) 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 X is complete if and only if Y is complete and Y
has the metric inherited from (X, d).
2. State and prove the Cantor's Intersection Theorem.
3. Define each of the following concepts for a metric space (X, d): (i)
A sequence converges in a metric space; (ii) A Cauchy sequence in X;
(iii) A complete metric space; (iv) Diameter of a subset A of X
Expert Solution
Step 1
We need to prove the following for the metric space :
(i) For is closed if and only if for each , there is an open set U containing x such that .
(ii) For if and only if there is a sequence such that .
(iii) If a sequence in X converges, it converges to a unique point.
(iv) Each and Every convergent sequence in X is a Cauchy sequence.
(v) A subset Y and X is complete if and only if Y is complete and Y has the metric inherited from (X, d).
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