topology exercice 13 part 4 5 6 Exercise 13on N (set of all nonnegative integers) byT = 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 T2. 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 Bis 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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Answered: 3. Show that every arithmetic…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.1: Infinite Sequences And Summation Notation

Problem 72E

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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.