Question

Consider the following statements:
1. Every compact Hausdorff space is normal.
2. Every metric space is normal.
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, recall key theorems connecting compactness, Hausdorff, and metric properties.

Answer 1

The Correct Option is A

Step 1: Understanding the properties of compact Hausdorff spaces. Every compact Hausdorff space is normal because compactness and the Hausdorff property ensure that disjoint closed sets can be separated by neighborhoods. Step 2: Understanding the properties of metric spaces. Every metric space is normal because metric spaces are paracompact and, consequently, normal. Step 3: Conclusion. Both statements are true. The correct answer is \( {(1)} \).