# 1. Complete Solution with Calculations ## Given - The set of integers \(\mathbb{Z}\) with the **Euclidean metric**: \( d(n, m) = |n - m| \). - Two balls in this metric space: 1. \( B(0, 1) = \{ n \in \mathbb{Z} : d(n, 0) < 1 \} \) 2. \( \overline{B}(0, 1) = \{ n \in \mathbb{Z} : d(n, 0) \leq 1 \} \) ### (A) **Describing the Balls** #### 1. \( B(0, 1) = \{ n \in \mathbb{Z} : |n| < 1 \} \) - The only integers satisfying \( |n| < 1 \) are \( n = 0 \). - **So,** \( B(0, 1) = \{ 0 \} \). #### 2. \( \overline{B}(0, 1) = \{ n \in \mathbb{Z} : |n| \leq 1 \} \) - The integers with \( |n| \leq 1 \) are \( n = -1, 0, 1 \). - **So,** \( \overline{B}(0, 1) = \{ -1, 0, 1 \} \). ### (B) **Open or Closed?** #### Open Set - A set \( U \subset \mathbb{Z} \) is **open** if, for every \( n \in U \), there is an \( \epsilon > 0 \) such that \( B(n, \epsilon) \subseteq U \). #### Closed Set - A set is **closed** if its complement is open. --- #### 1. \( B(0, 1) = \{0\} \) - For \( n = 0 \), \( B(0, 1) = \{0\} \). - For any \( \epsilon \leq 1 \), \( B(0, \epsilon) = \{0\} \). - So, \( \{0\} \) is open. - Complement: \( \mathbb{Z} \setminus \{0\} \) - For \( n \neq 0 \), \( B(n, 1) = \{n\} \subseteq \mathbb{Z} \setminus \{0\} \). - So, the complement is also open. - Therefore, \( \{0\} \) is **both open and closed** (clopen). #### 2. \( \overline{B}(0, 1) = \{-1, 0, 1\} \) - For \( n = 0 \), \( B(0, 1) = \{0\} \subseteq \{-1, 0, 1\} \). - For \( n = 1 \), \( B(1, 1) = \{1\} \subseteq \{-1, 0, 1\} \). - For \( n = -1 \), \( B(-1, 1) = \{-1\} \subseteq \{-1, 0, 1\} \). - So, \( \{-1, 0, 1\} \) is open. - Complement: \( \mathbb{Z} \setminus \{-1, 0, 1\} \) - For every \( n \notin \{-1, 0, 1\} \), \( B(n, 1) = \{n\} \subseteq \mathbb{Z} \setminus \{-1, 0, 1\} \). - So, the complement is also open. - Therefore, \( \{-1, 0, 1\} \) is **both open and closed** (clopen). --- # 2. Explanation of Each Part ### (A) Description of Balls - The **open ball** \( B(0, 1) \) includes all integers whose distance from 0 is less than 1. The only integer with absolute value less than 1 is 0. - The **closed ball** \( \overline{B}(0, 1) \) includes all integers whose distance from 0 is at most 1, i.e., -1, 0, and 1. ### (B) Openness and Closedness - In the discrete metric space (\(\mathbb{Z}\) with Euclidean metric), **every singleton set** is open because a sufficiently small radius ball around any point contains only that point. - Similarly, any **finite set** is open, and its complement is also open, making all finite sets both open and closed (clopen). --- # 3. Final Conclusion Statement - The open ball \( B(0, 1) \) in \((\mathbb{Z}, d)\) is \( \{0\} \), and the closed ball \( \overline{B}(0, 1) \) is \( \{-1, 0, 1\} \). - Both \( \{0\} \) and \( \{-1, 0, 1\} \) are **open and closed** (clopen) in \((\mathbb{Z}, d)\) with the Euclidean metric.