GATE 2025 MA – Question 53
f:ℝ²→ℝ² is $(e^{2\pi x}\cos2\pi y,e^{2\pi x}\sin2\pi y)$. Which statements hold?
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Correct answer: (A) Images of open sets are open.; (C) Images of dense sets are dense in ℝ².
Explanation
The Jacobian determinant is (2π)²e^(4πx)>0, so f is locally a homeomorphism and open, proving A. Its image is ℝ²∖{0}, not closed and not onto, refuting B,D by taking the whole domain. For dense G, continuity implies f(ℝ²)⊆closure(f(G)); since this punctured plane is dense, f(G) is dense. **A,C**.