Discussion
In the following, all subsets of Euclidean spaces are considered with the respective subspace topologies. Define an equivalence relation \(\sim\) on the sphere
\[ \begin{aligned} S = \{&(x_{1}, x_{2}, x_{3}) \in \mathbb{R}^{3} : \\ &x_{1}^{2} + x_{2}^{2} + x_{3}^{2} = 1\} \end{aligned} \]
by \((x_{1}, x_{2}, x_{3}) \sim (y_{1}, y_{2}, y_{3})\) if \(x_{3} = y_{3}\), for \((x_{1}, x_{2}, x_{3}), (y_{1}, y_{2}, y_{3}) \in S\). Let \([x_{1}, x_{2}, x_{3}]\) denote the equivalence class of \((x_{1}, x_{2}, x_{3})\), and let X denote the set of all such equivalence classes. Let \(\mathcal{L}: S \to X\) be given by \(\mathcal{L}((x_{1}, x_{2}, x_{3})) = [x_{1}, x_{2}, x_{3}]\). If X is provided with the quotient topology induced by the map \(\mathcal{L}\), then which one of the following is TRUE?
Answer: A
X is homeomorphic to the interval [-1, 1].
Two points of the sphere are identified exactly when they have the same height \(x_{3}\), so each equivalence class is a horizontal circle of latitude (a single point at each pole). The height map \(h(x_{1}, x_{2}, x_{3}) = x_{3}\) is continuous, onto [-1, 1], and constant precisely on the classes, so it induces a continuous bijection \(X \to [-1, 1]\).
A continuous bijection from a compact space (X is the image of the compact sphere) onto a Hausdorff space is a homeomorphism. The circle, disc and cylinder in the other options would keep more of the sphere's structure than the relation leaves, which collapses each latitude circle to a point.
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