The GATE Grind

GATE Topology: Countability and Separation – Previous Year Questions

8 GATE previous year questions on Countability and Separation (Topology, Mathematics) with answers and explanations, from every paper.

  1. GATE 2021 MA Q23 (1 mark, Multiple choice) – P:every compact metrizable topological space is separable. Q:every Hausdorff topology on a finite set is metrizable. Choose truth values.
  2. GATE 2022 MA Q53 (2 marks, Multiple select) – Let T be the smallest topology on ℝ for which every singleton is closed. For X=(ℝ,T), which hold?
  3. GATE 2022 MA Q54 (2 marks, Multiple select) – On ℤ let T be generated by A (m,n)=m+nk:k∈ℤ,m,n∈ℤ,n≠0. Which hold?
  4. GATE 2023 MA Q44 (2 marks, Multiple select) – On ℝ let τ=U⊆ℝ:U=∅ or 1∈U. Which statements hold?
  5. GATE 2024 MA Q17 (1 mark, Multiple choice) – I: Every compact Hausdorff space is normal. II: Every metric space is normal. Choose truth values.
  6. GATE 2024 MA Q50 (2 marks, Multiple select) – The intervals [a,b),a<b form a topology basis on ℝ. Which statements hold?
  7. GATE 2025 MA Q58 (2 marks, Multiple select) – X is uncountable with cocountable topology: ∅ is open and every set with finite or countable complement is open. Which statements hold?
  8. GATE 2026 MA Q47 (2 marks, Multiple choice) – A map f:X→Y is continuous and bijective. Which condition guarantees it is a homeomorphism?