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For n, ke Z, k0, we denote by A(n, k) the set of integers that are congruent to n modulo k: -2k+n, -k+n, n, k+n, 2k + n, A(n, k) = {... -3k+n, 3k+n, ...} CZ. We denote by T the collection of subsets U CZ with the property that: for every n EU, there exists a nonzero integer k such that A(n, k) CU. Please do the following: 1. prove that T is a topology on Z. 2. all the subsets A(n, k), with n,k € Z, k ‡ 0, are both open as well as closed in (Z, T).

For n, ke Z, k0, we denote by A(n, k) the set of integers that are congruent to n modulo k: -2k+n, -k+n, n, k+n, 2k + n, A(n, k) = {... -3k+n, 3k+n, ...} CZ. We denote by T the collection of subsets U CZ with the property that: for every n EU, there exists a nonzero integer k such that A(n, k) CU. Please do the following: 1. prove that T is a topology on Z. 2. all the subsets A(n, k), with n,k € Z, k ‡ 0, are both open as well as closed in (Z, T).

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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For n, k € Z, k ‡ 0, we denote by A(n, k) the set of integers that are congruent to n modulo k: A(n, k) = {... -k+n, n₂ k+n, 2k + n, 3k + n, -3k +n, -2k + n, ... } cz. We denote by T the collection of subsets UC Z with the property that: for every n € U, there exists a nonzero integer k such that A(n, k) CU. Please do the following: 1. prove that T is a topology on Z. 2. all the subsets A(n, k), with n, k = Z, k‡ 0, are both open as well as closed in (Z, T). 2
Transcribed Image Text:For n, k € Z, k ‡ 0, we denote by A(n, k) the set of integers that are congruent to n modulo k: A(n, k) = {... -k+n, n₂ k+n, 2k + n, 3k + n, -3k +n, -2k + n, ... } cz. We denote by T the collection of subsets UC Z with the property that: for every n € U, there exists a nonzero integer k such that A(n, k) CU. Please do the following: 1. prove that T is a topology on Z. 2. all the subsets A(n, k), with n, k = Z, k‡ 0, are both open as well as closed in (Z, T). 2
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