GATE 2022 MA – Question 22

Topology · Connectedness and Compactness · 1 mark · Multiple choice

Let T₁ be the usual subspace topology on [0,1] and T₂ any topology there. P:if T₁⊊T₂, then ([0,1],T₂) is not compact. Q:if T₂⊊T₁, then ([0,1],T₂) is not Hausdorff. Choose truth values.

P true, Q false
Both true
Both false
P false, Q true

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Show answer and explanation

Correct answer: (B) Both true

Explanation

A continuous bijection from a compact space to a Hausdorff space is a homeomorphism. If a strictly finer T₂ were compact, the identity to the usual Hausdorff interval would contradict strictness. If a strictly coarser T₂ were Hausdorff, the identity from the usual compact interval would do the same. **Both true, B**.

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