GATE 2024 MA – Question 17

Topology · Countability and Separation · 1 mark · Multiple choice

I: Every compact Hausdorff space is normal. II: Every metric space is normal. Choose truth values.

Both I and II true
I true, II false
I false, II true
Both false

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Correct answer: (A) Both I and II true

Explanation

Both standard separation theorems hold. In a compact Hausdorff space, finite subcovers turn pointwise disjoint neighborhoods into separating neighborhoods of disjoint closed sets. In a metric space, disjoint closed sets A,B can be separated by d(x,A)<d(x,B) and d(x,B)<d(x,A). **A**.

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