Exercise 10, § 5

Let W be the Warsaw circle, defined in Exercise 1, IV, p. 455, whose notations we take up again.

a) Let R be the equivalence relation in W whose only classes not reduced to a single element are B and S; let $p: W\rightarrow W/R$ be the canonical surjection. Prove that the quotient space $W/R$ is homotopic, but not homeomorphic, to the circle $\mathbf{S}_1$. Prove that the homomorphism $p_*$ is not surjective.

b) Let $R'$ be the equivalence relation in W of which $B\cup S$ is the only equivalence class not reduced to a single element; let $p': W\rightarrow W/R'$ be the canonical surjection. Prove that the space $W/R'$ is homeomorphic to $\mathbf{S}_1$, that the homomorphism $p_*$ is not surjective, but that the equivalence classes of $R'$ are connected.