1. Let \( \mathbb{Z} \) be the set of integers, and let \( \mathcal{B} \) be the set of all subsets of \( \mathbb{Z} \) of the form \(\{ n \}\) with \( n \) odd, or \(\{ n-1, n, n+1 \}\) with \( n \) even. (a) Show that \( \mathcal{B} \) is a basis for a topology on \( \mathbb{Z} \). (b) Is \( \mathbb{Z} \) equipped with this topology a Hausdorff space? (c) Is \( \mathbb{Z} \) equipped with this topology a connected space?

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Answered: 1. Let Z be the set of integers, and…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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1. Let Z be the set of integers, and let B be the set of all subsets of Z of the form {n} with n odd, or {n-1, n, n + 1} with n even. (a) Show that B is a basis for a topology on Z. (b) Is Z equipped with this topology a Hausdorff space? (c) Is Z equipped with this topology a connected space?