Mathematics · Topology

In the following, all subsets of Euclidean spaces are considered with the respective subspace topologies. Define an equivalence relation ~ on the sphere S = { (x1, x2, x3) ∈ ℝ3 : x12 + x22 + x32 = 1 } by (x1, x2, x3) ~ (y1, y2, y3) if x3 = y3, for (x1, x2, x3), (y1, y2, y3) ∈ S. Let [x1, x2, x3] denote the equivalence class of (x1, x2, x3), and let X denote the set of all such equivalence classes. Let L: S → X be given by L((x1, x2, x3)) = [x1, x2, x3]. If X is provided with the quotient topology induced by the map L, then which one of the following is TRUE?