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Exercise 13. Number theory VS topology: Furstenberg's proof Define a topology on N (set of all nonnegative integers) by TUUN, €, an arithmetic progression a+mN for some m > 1 and a+mNCU} 1. Verify that 7 defines a topology on N. 2. Verify that every nonempty open in 7 is infinite. 3. Show that every arithmetic progression is clopen in (N, 7).

Exercise 13. Number theory VS topology: Furstenberg's proof Define a topology on N (set of all nonnegative integers) by TUUN, €, an arithmetic progression a+mN for some m > 1 and a+mNCU} 1. Verify that 7 defines a topology on N. 2. Verify that every nonempty open in 7 is infinite. 3. Show that every arithmetic progression is clopen in (N, 7).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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Exercise 13. Number theory VS topology: Furstenberg's proof Define a topology T on N (set of all nonnegative integers) by T = OU{UCN, Va EU, an arithmetic progression a+mN for some m > 1 and a+mN CU} 1. Verify that defines a topology on N. 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. Number theory VS topology: Furstenberg's proof Define a topology T on N (set of all nonnegative integers) by T = OU{UCN, Va EU, an arithmetic progression a+mN for some m > 1 and a+mN CU} 1. Verify that defines a topology on N. 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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