Exercise 8, § 5

Let X be the quotient space of the space $\mathbf{R}\times \{0,1\}$ by the least fine equivalence relation which identifies $(x,0)$ and $(x,1)$ for every $x\in \mathbf{R}^*$ (“right numerical line whose origin is doubled”).

(a) Prove that $\pi_1(X, x)$ is isomorphic to $\mathbf{Z}$, for every point $x\in X$.

(b) Let $X^{(2)}$ be the quotient space of the space $X^2$ by the operation of the symmetric group $\mathfrak{S}_2$ on the space $X^2$ acting by permutation of the coordinates. Prove that $\pi_1(X^{(2)}, y) = 0$ for every $y\in X^{(2)}$.