On the unit sphere S⊂ℝ³ identify points exactly when their third coordinates agree. Give the equivalence-class space the quotient topology. To which space is it homeomorphic?
Height h(x)=x₃ is continuous and onto [−1,1], with fibers exactly the equivalence classes. It induces a continuous bijection from the compact quotient to the Hausdorff interval, hence a homeomorphism. **A**.
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