Correct answer: (C) U is connected but V disconnected.
Explanation
Removing a countable set from the plane leaves it path connected: between any two remaining points choose an intermediate point avoiding the countably many lines that would make a joining segment hit a deleted point. Thus U is connected. The removed positive hyperbola separates V into the region $x>0,y>1/x$ and its complementary side, two nonempty relatively open parts. Hence V is disconnected and C holds.
More Connectedness and Compactness questions
GATE 2020 MA Q40– Equip the power set P(R) with d(U,V)= x R U(x)- V(x) . The set \\m\:m Z\ is
GATE 2020 MA Q45– With Euclidean induced metrics, let X=\(0,0)\\(x,(1/x)):x0\ and Y=\(0,0)\\(x,x(1/x)):x0\.…
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…
GATE 2026 MA Q49– S= n=1\(x,y) R2:(x-n)2+y2=1/n2\ with the usual topology. Which statements hold?
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…