Proof that the Klein bottle is homeomorphic to $T/S$ where $T$ is the torus of revolution and $S$ is the equivalence relation given by $(x, y, z) \sim (x', y', z')$ if and only if $(x, y, z) = \pm (x', y', z')$. How do I must do it? Sorry, I am new in this.
2026-03-25 23:37:24.1774481844
Homeomorphism of Klein Bottle
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I could use this lemma: Suppose X is a topological space, ∼ an equivalence relation on X, and Q = X/ ∼ the quotient space (i.e. the set of equivalence classes, with the quotient topology). Suppose Y is a space and f : X → Y is a continuous surjective map such that the equivalence classes in X are precisely the point-inverses under f; that is,foreachx,y ∈ X,x∼y ⇐⇒ f(x)=f(y). If X is compact and Y is Hausdorff, then X/ ∼ is homeomorphic to Y . Right?