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?
Consider the following statements:
1. Every compact Hausdorff space is normal.
2. Every metric space is normal.
Which one of the following is correct?
1. Every compact Hausdorff space is normal.
2. Every metric space is normal.
Which one of the following is correct?
Show Hint
For topology questions, recall key theorems connecting compactness, Hausdorff, and metric properties.
Updated On: Feb 1, 2025
- Both I and II are TRUE
- I is TRUE and II is FALSE
- I is FALSE and II is TRUE
- Both I and II are FALSE
Show Solution
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The Correct Option is A
Solution and Explanation
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 \( {(1)} \).
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