4. Let B be a collection of all arithmetic progression of positive integers. Show that B is a basis for T.
4. Let B be a collection of all arithmetic progression of positive integers. Show that B is a basis for T.
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topology exeecice 13 part 4 5 6
![**Exercise 13: Number Theory vs. Topology: Furstenberg's Proof**
Define a topology \( \tau \) on \( \mathbb{N} \) (set of all non-negative integers) by
\[ \tau = \emptyset \cup \{ U \subseteq \mathbb{N}, \forall a \in U, \exists \text{ an arithmetic progression } a + m\mathbb{N} \text{ for some } m \geq 1 \text{ and } a + m\mathbb{N} \subset U \} \]
1. **Verify that \( \tau \) defines a topology on \( \mathbb{N} \).**
2. **Verify that every nonempty open set in \( \tau \) is infinite.**
3. **Show that every arithmetic progression is clopen in \( (\mathbb{N}, \tau) \).**
4. **Let \( \mathcal{B} \) be a collection of all arithmetic progressions of positive integers. Show that \( \mathcal{B} \) is a basis for \( \tau \).**
5. **Show that \( (\mathbb{N}, \tau) \) is Hausdorff.**
6. **Prove that for every prime number \( p \), the set \(\{ np; \, n \geq 1 \}\) is closed.**
7. **Prove that there are infinitely many prime numbers.**
---
**Explanation of the Concepts:**
- **Topology \( \tau \):** A collection of subsets of a set \( \mathbb{N} \) that satisfies certain conditions. Here, it involves arithmetic progressions.
- **Clopen Sets:** Sets that are both open and closed in a given topology.
- **Hausdorff Space:** A space in which any two distinct points have disjoint neighborhoods.
- **Basis for a Topology:** A collection of open sets such that every open set can be written as a union of elements from the basis.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3b5f7db1-a103-4627-9a7f-8775edb35b3d%2F26008aa0-77db-4fd8-92ce-d30db96ccc62%2Fmju0fne_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Exercise 13: Number Theory vs. Topology: Furstenberg's Proof**
Define a topology \( \tau \) on \( \mathbb{N} \) (set of all non-negative integers) by
\[ \tau = \emptyset \cup \{ U \subseteq \mathbb{N}, \forall a \in U, \exists \text{ an arithmetic progression } a + m\mathbb{N} \text{ for some } m \geq 1 \text{ and } a + m\mathbb{N} \subset U \} \]
1. **Verify that \( \tau \) defines a topology on \( \mathbb{N} \).**
2. **Verify that every nonempty open set in \( \tau \) is infinite.**
3. **Show that every arithmetic progression is clopen in \( (\mathbb{N}, \tau) \).**
4. **Let \( \mathcal{B} \) be a collection of all arithmetic progressions of positive integers. Show that \( \mathcal{B} \) is a basis for \( \tau \).**
5. **Show that \( (\mathbb{N}, \tau) \) is Hausdorff.**
6. **Prove that for every prime number \( p \), the set \(\{ np; \, n \geq 1 \}\) is closed.**
7. **Prove that there are infinitely many prime numbers.**
---
**Explanation of the Concepts:**
- **Topology \( \tau \):** A collection of subsets of a set \( \mathbb{N} \) that satisfies certain conditions. Here, it involves arithmetic progressions.
- **Clopen Sets:** Sets that are both open and closed in a given topology.
- **Hausdorff Space:** A space in which any two distinct points have disjoint neighborhoods.
- **Basis for a Topology:** A collection of open sets such that every open set can be written as a union of elements from the basis.
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