Let ℝ3 be a topological space with the usual topology and ℚ denote the set of rational numbers. Define the subspaces 𝑋, 𝑌, 𝑍 and 𝑊 of ℝ3 as follows:
𝑋 = {(𝑥, 𝑦, 𝑧) ∈ ℝ3 ∶ |𝑥| + |𝑦| + |𝑧| ∈ ℚ}
𝑌 = {(𝑥, 𝑦, 𝑧) ∈ ℝ3 ∶ 𝑥𝑦𝑧 = 1}
𝑍 = {(𝑥, 𝑦, 𝑧) ∈ ℝ3 ∶ 𝑥2 + 𝑦2 + 𝑧2 = 1}
𝑊 = {(𝑥, 𝑦, 𝑧) ∈ ℝ3 ∶ 𝑥𝑦𝑧 = 0 }
Which of the following statements is correct?
1
𝑋 is homeomorphic to Y
2
𝑍 is homeomorphic to W
3
𝑌 is homeomorphic to W
4
𝑋 is NOT homeomorphic to W
5
Question Not Attempted