Exercise 5, § 1

Let $X$ be the subspace of $\mathbf{R}^2$ consisting of the union of $[0,1]\times \{0,1\}$, of $\{0\} \times [0,1]$ and of the segments $\{\frac{1}{n}\} \times [0,1]$, for $n\geqslant 1$.

(a) Prove that every point $a$ of X admits a neighbourhood V such that the image of $\pi_1(V, a)$ in $\pi_1(X, a)$ is trivial.

(b) Prove that the quotient topology of the topology of compact convergence on the group $\pi_1(X, a)$ is not discrete.