GATE Topology: Connectedness and Compactness – Previous Year Questions
15 GATE previous year questions on Connectedness and Compactness (Topology, Mathematics) with answers and explanations, from every paper.
- GATE 2021 MA Q52 – Let X₁ be ℝ with upper-limit basis (a,b], and X₂ be ℝ with the cofinite topology. Which holds?
- GATE 2022 MA Q17 – In usual ℝ² let A=x²+y²≤1,B=(x−2)²+y²≤1, M=A∪B and N=int A∪int B. Which holds?
- GATE 2022 MA Q22 – 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…
- GATE 2022 MA Q24 – Give Y=∏ₙ₌₁∞ℝ its product topology and Δ=(x,x,x,…):x∈ℝ its subspace topology. Which holds?
- GATE 2023 MA Q12 – In usual ℝ³ let X= x + y + z ∈ℚ, Y=xyz=1, Z=x²+y²+z²=1, W=xyz=0. Which statement is correct?
- GATE 2023 MA Q18 – In usual ℝ², X=(x,x sin(1/x)):0<x≤1∪(0×ℝ) and Y=(x,sin(1/x)):0<x≤1∪(0×ℝ). P:X is connected. Q:Y is connected. Choose truth values.
- GATE 2023 MA Q24 – A continuous map is proper when inverse images of compact sets are compact. Using usual/subspace/product topologies, which map is proper?
- GATE 2024 MA Q18 – On ℤ give the topology with basis S(a,b)=an+b:n∈ℤ, a≠0. I: Every S(a,b) is both open and closed. II: The only connected set containing x is x. Choose…
- GATE 2024 MA Q41 – X=ℝ/ℤ has the quotient topology from usual ℝ. I:X is compact. II:Removing any one point leaves X connected. Choose truth values.
- GATE 2025 MA Q15 – In ℝ⁴ define S by x₁²+x₂²+x₃²−x₄²=0, T by the same expression=1, U by it=−1. Which statement holds?
- GATE 2025 MA Q22 – f(x₁,x₂)=(x₁−x₂)² and g(x₁,x₂)=2x₁²+2x₂²−x₁x₂. S1: f⁻¹(K) is compact for every compact K⊂ℝ. S2: g⁻¹(K) is compact for every such K. Choose the truth…
- GATE 2025 MA Q57 – A nonconstant continuous f:S¹→ℝ is defined on the unit circle with usual topology. Which statements hold?
- GATE 2026 MA Q24 – Give ℝ the cofinite topology =\A R:Ac is finite\\\. Which statements hold?
- GATE 2026 MA Q45 – For subsets E₁,E₂ of a normed space, define E₁+E₂ = x+y:x∈E₁,y∈E₂ and E₁×E₂ as their Cartesian product. Which statement is NOT true?
- GATE 2026 MA Q49 – S= n=1\(x,y) R2:(x-n)2+y2=1/n2\ with the usual topology. Which statements hold?