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3. Show that every arithmetic progression is clopen in (N, 7). 4. Let B be a collection of all arithmetic progression of positive integers. Show that B is a basis for T. 5. Show that (N, 7) is Hausdorff.

3. Show that every arithmetic progression is clopen in (N, 7). 4. Let B be a collection of all arithmetic progression of positive integers. Show that B is a basis for T. 5. Show that (N, 7) is Hausdorff.

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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topology exercice 13 part 4 5 6

Exercise 13 on N (set of all nonnegative integers) by T = OU{UCN, Va EU, an arithmetic progression a+mN for some m > 1 and a+mNCU} 1. Verify that defines a topology on N. |Number theory VS topology: Furstenberg's proof Define a topology T 2. Verify that every nonempty open in 7 is infinite. 3. Show that every arithmetic progression is clopen in (N, 7). 4. Let B be a collection of all arithmetic progression of positive integers. Show that B is a basis for 7. 5. Show that (N, ) is Hausdorff. 6. Prove that for every prime number p, the set {np; n 1} is closed. 7. Prove that there are infinitely many primes numbers.
Transcribed Image Text:Exercise 13 on N (set of all nonnegative integers) by T = OU{UCN, Va EU, an arithmetic progression a+mN for some m > 1 and a+mNCU} 1. Verify that defines a topology on N. |Number theory VS topology: Furstenberg's proof Define a topology T 2. Verify that every nonempty open in 7 is infinite. 3. Show that every arithmetic progression is clopen in (N, 7). 4. Let B be a collection of all arithmetic progression of positive integers. Show that B is a basis for 7. 5. Show that (N, ) is Hausdorff. 6. Prove that for every prime number p, the set {np; n 1} is closed. 7. Prove that there are infinitely many primes numbers.
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