Exercise 13. Number theory VS topology: Furstenberg's proof Define a topology on N (set of all nonnegative integers) by TUUN, €, an arithmetic progression a+mN for some m > 1 and a+mNCU} 1. Verify that 7 defines a topology on N. 2. Verify that every nonempty open in 7 is infinite. 3. Show that every arithmetic progression is clopen in (N, 7).
Exercise 13. Number theory VS topology: Furstenberg's proof Define a topology on N (set of all nonnegative integers) by TUUN, €, an arithmetic progression a+mN for some m > 1 and a+mNCU} 1. Verify that 7 defines a topology on N. 2. Verify that every nonempty open in 7 is infinite. 3. Show that every arithmetic progression is clopen in (N, 7).
Advanced Engineering Mathematics
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Author:Erwin Kreyszig
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topology part 1 2 3
![**Exercise 13: Number Theory vs Topology: Furstenberg's Proof**
Define a topology \( \tau \) on \( \mathbb{N} \) (set of all nonnegative integers) by
\[
\tau = \emptyset \cup \{U \subseteq \mathbb{N}, \forall a \in U, \exists \, \text{an arithmetic progression} \, a + m \mathbb{N} \, \text{for some} \, m \geq 1 \, \text{and} \, a + m \mathbb{N} \subseteq U\}
\]
1. Verify that \( \tau \) defines a topology on \( \mathbb{N} \).
2. Verify that every nonempty open 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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbc44ee4c-f0ee-41b9-a7bf-13469d6ba446%2F736fb5c9-ee30-4a17-9d07-9cf0285bb91b%2F73sw045_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 nonnegative integers) by
\[
\tau = \emptyset \cup \{U \subseteq \mathbb{N}, \forall a \in U, \exists \, \text{an arithmetic progression} \, a + m \mathbb{N} \, \text{for some} \, m \geq 1 \, \text{and} \, a + m \mathbb{N} \subseteq U\}
\]
1. Verify that \( \tau \) defines a topology on \( \mathbb{N} \).
2. Verify that every nonempty open 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.
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