1. (X, d) is a metric space and A and B are subsets of X. Prove each of the following if the boundary of a set A, denoted as bd(A), is defined as bd(A) = An (X-A). Prove each of the following: (i) bd(A) is a closed set. (ii) A set A is closed if and only if bd (A) CA. (iii) clA (iv) If the metric d on X is the discrete metric, discuss the boundary of a set AC X. = Aº U bdA. 3. If G is an open dense subset of X, then X-G is nowhere dense, where G is a non-empty proper subset of a metric space X. 4. Let U, V be subsets of a metric space X and V be open. Show that if UnV = 0, then clUV = 0. " 5. Give an example of a family {Un new} of open subsets of R with usual metric, such that new Un is not open.
1. (X, d) is a metric space and A and B are subsets of X. Prove each of the following if the boundary of a set A, denoted as bd(A), is defined as bd(A) = An (X-A). Prove each of the following: (i) bd(A) is a closed set. (ii) A set A is closed if and only if bd (A) CA. (iii) clA (iv) If the metric d on X is the discrete metric, discuss the boundary of a set AC X. = Aº U bdA. 3. If G is an open dense subset of X, then X-G is nowhere dense, where G is a non-empty proper subset of a metric space X. 4. Let U, V be subsets of a metric space X and V be open. Show that if UnV = 0, then clUV = 0. " 5. Give an example of a family {Un new} of open subsets of R with usual metric, such that new Un is not open.
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. (X, d) is a metric space and A and B are subsets of X. Prove each
of the following if the boundary of a set A, denoted as bd(A), is defined
as bd(A) = An (X - A). Prove each of the following:
(i) bd(A) is a closed set.
(ii) A set A is closed if and only if bd (A) CA.
(iii) clA = Aº U bd.A.
(iv) If the metric d on X is the discrete metric, discuss the boundary
of a set AC X.
3. If G is an open dense subset of X, then X - G is nowhere dense,
where G is a non-empty proper subset of a metric space X.
4. Let U, V be subsets of a metric space X and V be open. Show that
if Un V = 0, then clUnV = 0.
"
5. Give an example of a family {Un new} of open subsets of R
with usual metric, such that new Un is not open.
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