Question

Consider the topology on \( {Z} \) with basis \( S(a,b) = \{an + b : n \in {Z\} \), where \( a, b \in {Z} \) and \( a \neq 0 \). Consider the following statements:} 1. \( S(a, b) \) is both open and closed for each \( a, b \in {Z} \) with \( a \neq 0 \).
2. The only connected set containing \( x \in {Z} \) is \( \{x\} \).
Which one of the following is correct?

• A. Both I and II are TRUE
• B. I is TRUE and II is FALSE
• C. I is FALSE and II is TRUE
• D. Both I and II are FALSE

Hint:

For topology questions on connectedness, analyze the structure of basic open sets and their complements.

Answer 1

The Correct Option is A

Step 1: Checking whether \( S(a, b) \) is open and closed. In the given topology, each \( S(a, b) \) is a basic open set and also its own complement, making it both open and closed. Step 2: Verifying connected sets. A connected set in this topology cannot contain more than one element because the space \( {Z} \) is totally disconnected in this topology. Step 3: Conclusion. Both statements are true. The correct answer is \( {(1)} \).