Question

Let \( X \) be the space \( {R}/{Z} \) with the quotient topology induced from the usual topology on \( {R} \). Consider the following statements:
1. \( X \) is compact.
2. \( X \setminus \{z\} \) is connected for any \( z \in X \).
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 quotient topologies, consider properties like compactness and connectedness inherited from the original space.

Answer 1

The Correct Option is A

Step 1: Compactness of \( X \). The space \( {R}/{Z} \) is compact because \( {R} \) is locally compact, and the quotient by \( {Z} \) identifies points separated by integers, creating a compact topology. Step 2: Connectedness of \( X \setminus \{z\} \). Removing a point from \( {R}/{Z} \) does not disconnect the space, as \( {R}/{Z} \) is homeomorphic to a circle, and a circle remains connected after the removal of a single point. Step 3: Conclusion. Both statements are true. The correct answer is \( {(1)} \).