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Definition: A subset E of a metric space (X, d) is called dicconnected if there exist subsets A and B of X which satisfy the following properties: (i) E = AUB. (ii) A ‡ Ø and B ‡ Ø. (iii) An B = 0 and BnA = Ø. If E is not disconnected, then it is called a connected set. 1. Give an example of a connected and an example of a disconnected set in R with usual (Euclidean) metric. In each case, justify your claim. 2. Prove that an interval in R is a connected set.

Definition: A subset E of a metric space (X, d) is called dicconnected if there exist subsets A and B of X which satisfy the following properties: (i) E = AUB. (ii) A ‡ Ø and B ‡ Ø. (iii) An B = 0 and BnA = Ø. If E is not disconnected, then it is called a connected set. 1. Give an example of a connected and an example of a disconnected set in R with usual (Euclidean) metric. In each case, justify your claim. 2. Prove that an interval in R is a connected set.

Advanced Engineering Mathematics
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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Definition: A subset E of a metric space (X, d) is called dicconnected if there exist subsets A and B of X which satisfy the following properties: (i) E = AUB. (ii) A ‡ Ø and B ‡ Ø. (iii) AnB = 0 and B₁Ā = 0. If E is not disconnected, then it is called a connected set. 1. Give an example of a connected and an example of a disconnected set in R with usual (Euclidean) metric. In each case, justify your claim. 2. Prove that an interval in R is a connected set.
Transcribed Image Text:Definition: A subset E of a metric space (X, d) is called dicconnected if there exist subsets A and B of X which satisfy the following properties: (i) E = AUB. (ii) A ‡ Ø and B ‡ Ø. (iii) AnB = 0 and B₁Ā = 0. If E is not disconnected, then it is called a connected set. 1. Give an example of a connected and an example of a disconnected set in R with usual (Euclidean) metric. In each case, justify your claim. 2. Prove that an interval in R is a connected set.
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