Exercise 8, § 5

We take up again the notations of Exercises 6 and 7; let also $p: \Omega_0(P)\rightarrow$ $\pi_1(P,0)$ be the canonical surjection.

(a) For every pair $(m, n)$ of integers $\geqslant$ 1 such that $m=\not n$, put $\lambda_{m,n}= (\alpha_m\alpha_n\alpha^{-1}_m\alpha^{-1}_n)^{m+n}$ and $\lambda_{m,n}= (\alpha_1\alpha_m\alpha^{-1}_1\alpha^{-1}_m)^n$; let S be the set of pairs of the form $(\lambda_{m,n}, \mu_{m,n})$ in $\pi_1(P,0)\times \pi_1(P,0)$. Prove that S is not closed but that its inverse image in $\Omega_0(P)\times \Omega_0(P)$ is closed. Deduce that the mapping $p\times p: \Omega_0(P)^2\times \pi_1(P,0)^2$ is not strict.

(b) Let T be the set of classes of loops of the form $\lambda_{m,n}\mu_{m,n}$ in $\pi_1(P,0)$. Prove that T is closed.

(c) Prove that the law of composition of $\pi_1(P,0)$ is not continuous when the group $\pi_1(P,0)$ is endowed with the quotient topology of compact convergence.[^1]

[^1]: This example is due to P. Fabel, “Multiplication is discontinuous in the Hawaiian earring group (with the quotient topology)”, Bull. Polish Acad. Sci. Math. 59 (2011), p. 77-83.