The GATE Grind

GATE 2026 MA – Question 24

Topology · Connectedness and Compactness · 1 mark · Multiple select

Give ℝ the cofinite topology $\tau=\{A\subseteq\mathbb R:A^c\text{ is finite}\}\cup\{\varnothing\}$. Which statements hold?

  1. It is Hausdorff.
  2. Every finite subset is closed.
  3. It is compact.
  4. It is connected.

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Show answer and explanation

Correct answer: (B) Every finite subset is closed.; (C) It is compact.; (D) It is connected.

Explanation

Complements of finite sets are open, proving B. From any cover choose a nonempty open set; only finitely many points remain, so finitely many other cover members suffice, proving C. Any two nonempty opens intersect because their complements are finite, proving connectedness D and refuting Hausdorffness A. **B,C,D**.

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