GATE 2023 MA – Question 24

Topology · Connectedness and Compactness · 1 mark · Multiple choice

A continuous map is proper when inverse images of compact sets are compact. Using usual/subspace/product topologies, which map is proper?

ℝ∖{0}→ℝ, f(x)=x
ℝ²→ℝ², f(x,y)=(x+y,y)
ℝ²→ℝ, f(x,y)=x
ℝ²→ℝ, f(x,y)=x²−y²

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Show answer and explanation

Correct answer: (B) ℝ²→ℝ², f(x,y)=(x+y,y)

Explanation

The shear in B is a homeomorphism with continuous inverse (u,v)↦(u−v,v); inverse images of compact sets are its inverse’s continuous images, hence compact. A fails on [−1,1], whose punctured preimage is noncompact. C and D fail because the preimage of {0} is unbounded. **B**.

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