Exercise 6, § 5

For every integer $n\geqslant 1$, put $x_n= (1/n,0)$ and denote by $C_n$ the circle of the numerical plane having diameter $(0, x_n)$. Let P be the union of the spaces $C_n$, for $n\in \mathbf{N}$ (“Hawaiian earring”, cf. I, p. 146, exerc. 5); for every integer $n\geqslant 1$, denote by $P_n$ the union of the spaces $C_m$, for $1\leqslant m\leqslant n$ and by $P^n$ the union of the spaces $C_m$, for $m > n$.

a) Let $n$ be an integer $\geqslant 1$ and let $r_n$ be the mapping of P into $P_n$ which maps a point $x\in P$ onto itself if $x\in P_n$, and onto 0 otherwise. Prove that $r_n$ is a continuous retraction of the canonical injection of $P_n$ into P.

b) Prove that $P_n$ is locally contractible. Deduce that every point of P distinct from the origin has a contractible neighbourhood at this point.

c) Prove that the canonical inclusions of $P_n$ and $P^n$ into P induce a group isomorphism of the free product $\pi_1(P_n,0)*\pi_1(P^n,0)$ onto $\pi_1(P,0)$. (Prove that the canonical injection of $P^n$ into the complement $U_n$ of the set $\{x_1, . . . , x_n\}$ is a homotopy equivalence. Apply then Proposition 2 of IV, p. 422 to the covering $(P_n,U_n)$ of P.) d) Let $\gamma \in \pi_1(P,0)$ be a class of loops such that $(r_n)_*(\gamma ) =\varepsilon_0$ for every integer $n\geqslant 1$; prove that $\gamma =\varepsilon_0$.

e) The canonical homomorphism of $\pi_1(P,0)$ into lim$\leftarrow -_n\pi_1(P_n,0)$ deduced from the family $((r_n)_*)$ is injective and the admissible topology on $\pi_1(P,0)$ is the inverse image of the topology of the projective limit on lim$\leftarrow -_n\pi_1(P_n,0)$, the groups $\pi_1(P_n,0)$ being provided with the discrete topology.

f) Prove that the path-connected components of $\Omega_0(P)$ are closed, as are those of $\Omega_{(0,0)}(P\times P)$.