Let 𝑋 and 𝑌 be two topological spaces. A continuous map 𝑓 ∶ 𝑋 → 𝑌 is said to be proper if 𝑓−1 (𝐾) is compact in 𝑋 for every compact subset 𝐾 of 𝑌, where 𝑓−1 (𝐾) is defined by 𝑓−1 (𝐾) = {𝑥 ∈ 𝑋 ∶ 𝑓(𝑥) ∈ 𝐾}.
Consider ℝ with the usual topology. If ℝ ∖ {0} has the subspace topology induced from ℝ and ℝ × ℝ has the product topology, then which of the following maps is proper?
1
𝑓: ℝ ∖ {0} → ℝ defined by 𝑓(𝑥) = x
2
𝑓: ℝ × ℝ → ℝ × ℝ defined by 𝑓(𝑥, 𝑦) = (𝑥 + 𝑦, 𝑦)
3
𝑓: ℝ × ℝ → ℝ defined by 𝑓(𝑥, 𝑦) = x
4
𝑓: ℝ × ℝ → ℝ defined by 𝑓(𝑥, 𝑦) = 𝑥2 − 𝑦2
5
Question Not Attempted