Question:
Consider \( (\mathbb{Z}, T) \), where \( T \) is the topology generated by sets of the form
\[
A_{m,n} = \{ m + nk \mid k \in \mathbb{Z} \}, \quad \text{for } m, n \in \mathbb{Z} \text{ and } n \neq 0.
\]
Then, which of the following statements are TRUE?
Consider \( (\mathbb{Z}, T) \), where \( T \) is the topology generated by sets of the form
\[
A_{m,n} = \{ m + nk \mid k \in \mathbb{Z} \}, \quad \text{for } m, n \in \mathbb{Z} \text{ and } n \neq 0.
\]
Then, which of the following statements are TRUE?
Show Hint
In topological spaces, verify connectedness, Hausdorff condition, and metrizability by examining the properties of open sets and the behavior of points under the topology.
Updated On: Dec 4, 2025
- \( (\mathbb{Z}, T) \) is connected
- Each \( A_{m,n} \) is a closed subset of \( (\mathbb{Z}, T) \)
- \( (\mathbb{Z}, T) \) is Hausdorff
- \( (\mathbb{Z}, T) \) is metrizable
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The Correct Option is B, C, D
Solution and Explanation
Step 1: Analyzing Statement (A).
The topology \( T \) is generated by sets of the form \( A_{m,n} = \{ m + nk \mid k \in \mathbb{Z} \} \), where \( m \) and \( n \) are integers, and \( n \neq 0 \). These sets are not connected because they consist of individual equivalence classes modulo \( n \), and it is possible to separate them into disjoint open sets. Hence, the space \( (\mathbb{Z}, T) \) is not connected. Therefore, statement (A) is false.
Step 2: Analyzing Statement (B).
Each set \( A_{m,n} \) is of the form \( \{ m + nk \mid k \in \mathbb{Z} \} \), which is a set of points in \( \mathbb{Z} \) separated by multiples of \( n \). Since \( A_{m,n} \) is a finite set of points, it is closed in the topology \( T \), because it is generated by the open sets in \( T \). Therefore, statement (B) is true.
Step 3: Analyzing Statement (C).
To check if \( (\mathbb{Z}, T) \) is Hausdorff, we need to check if for any two distinct points, there exist disjoint open sets separating them. In the topology \( T \), we can always find such disjoint sets by considering the sets \( A_{m,n} \), which separate distinct points of \( \mathbb{Z} \). Hence, \( (\mathbb{Z}, T) \) is Hausdorff, so statement (C) is true.
Step 4: Analyzing Statement (D).
A space is metrizable if there exists a metric that induces the topology. Since the topology \( T \) on \( \mathbb{Z} \) is generated by a countable basis of open sets, it is metrizable. Specifically, we can define a metric based on the separation of points by multiples of integers, making \( (\mathbb{Z}, T) \) metrizable. Therefore, statement (D) is true.
Step 5: Final Answer.
The correct answer is (B), (C), (D) because these statements are true.
The topology \( T \) is generated by sets of the form \( A_{m,n} = \{ m + nk \mid k \in \mathbb{Z} \} \), where \( m \) and \( n \) are integers, and \( n \neq 0 \). These sets are not connected because they consist of individual equivalence classes modulo \( n \), and it is possible to separate them into disjoint open sets. Hence, the space \( (\mathbb{Z}, T) \) is not connected. Therefore, statement (A) is false.
Step 2: Analyzing Statement (B).
Each set \( A_{m,n} \) is of the form \( \{ m + nk \mid k \in \mathbb{Z} \} \), which is a set of points in \( \mathbb{Z} \) separated by multiples of \( n \). Since \( A_{m,n} \) is a finite set of points, it is closed in the topology \( T \), because it is generated by the open sets in \( T \). Therefore, statement (B) is true.
Step 3: Analyzing Statement (C).
To check if \( (\mathbb{Z}, T) \) is Hausdorff, we need to check if for any two distinct points, there exist disjoint open sets separating them. In the topology \( T \), we can always find such disjoint sets by considering the sets \( A_{m,n} \), which separate distinct points of \( \mathbb{Z} \). Hence, \( (\mathbb{Z}, T) \) is Hausdorff, so statement (C) is true.
Step 4: Analyzing Statement (D).
A space is metrizable if there exists a metric that induces the topology. Since the topology \( T \) on \( \mathbb{Z} \) is generated by a countable basis of open sets, it is metrizable. Specifically, we can define a metric based on the separation of points by multiples of integers, making \( (\mathbb{Z}, T) \) metrizable. Therefore, statement (D) is true.
Step 5: Final Answer.
The correct answer is (B), (C), (D) because these statements are true.
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