Exercise 9, § 5

Let X be a topological space, let R be an equivalence relation in X and let $p: X\rightarrow X/R$ be the canonical surjection. Let $x\in X$.

(a) Suppose that X is connected and locally arcwise connected, that the equivalence classes are connected sets, and that the space $X/R$ is unravelable. Prove that the homomorphism $p_*:\pi_1(X, x)\rightarrow \pi_1(X/R, p(x))$ is surjective.

(b) Suppose that $X =\mathbf{I}$ and that $\{0,1\}$ is the only equivalence class which is not reduced to an element. Prove that the homomorphism $p_*$ is not surjective.