11. Let (X, d) be a pseudometric space, and suppose d has the property that d(a, b) >0 whenever ab. Prove that every finite subset of X is closed.
11. Let (X, d) be a pseudometric space, and suppose d has the property that d(a, b) >0 whenever ab. Prove that every finite subset of X is closed.
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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Question
Basic Pseudometric Spaces
![11. Let (X. d) be a pseudometric space, and suppose d has the property that
d(a, b) > 0 whenever a b. Prove that every finite subset of X is closed.
12. Let (X, d) be the space of real numbers with the usual pseudometric. For each
positive integer n, let An= (1/n, 1]. Find cl(U{A}), and find U{cl(An)}.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F79599c56-a340-49a0-b0ff-829b3947a798%2F1a14b317-ea23-4997-87d2-f9f25b6f61d4%2F5i2q8ro_processed.jpeg&w=3840&q=75)
Transcribed Image Text:11. Let (X. d) be a pseudometric space, and suppose d has the property that
d(a, b) > 0 whenever a b. Prove that every finite subset of X is closed.
12. Let (X, d) be the space of real numbers with the usual pseudometric. For each
positive integer n, let An= (1/n, 1]. Find cl(U{A}), and find U{cl(An)}.
Expert Solution
Step 1
11)
To prove that every finite subset of X is closed, we need to show that its complement is open. That is, for any point x in the complement of the finite subset, we need to find an open ball centered at x that is entirely contained in the complement.
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