Exercise 4, § 1

For every integer $n\geqslant 1$, let $X_n$ be the union of the three segments of the numerical plane $\mathbf{R}^2$ whose endpoints are $(0,1)$, $(1/2n,0)$ and $(1/(2n-1),0)$. Let $X_0$ be the segment with endpoints $(0,1)$ and $(0,0)$. Let X be the union of the sets $X_n$, for $n\geqslant 1$, and let Y = X. Let $a= (0,1)$.

(a) Exercise prove that $Y = X\cup X_0$.

(b) Exercise prove that X is locally contractile.

(c) Exercise prove that the canonical injection of X into Y induces a group isomorphism of $\pi_1(X, a)$ onto $\pi_1(Y, a)$.

(d) Exercise prove that the quotient topology of compact convergence on $\pi_1(Y, a)$ is not discrete.