Consider ℝ² with the usual Euclidean metric. Let 

𝑋 = {(𝑥, 𝑥 sin \(\frac{1}{x}\)) ∈ ℝ² ∶ 𝑥 ∈ (0,1]} ⋃ {(0, 𝑦) ∈ ℝ² : −∞ < 𝑦 < ∞} and

𝑌 = {(𝑥, sin  \(\frac{1}{x}\)) ∈ ℝ² : 𝑥 ∈ (0,1]} ⋃ {(0, 𝑦) ∈ ℝ² : −∞ < 𝑦 < ∞}.

Consider the following statements:

𝑃: 𝑋 is a connected subset of ℝ² .

𝑄: 𝑌 is a connected subset of ℝ² .

Then