Remark

Let $(X, e)$ be a Hopf space. By Prop. 3, the mapping of composition of path classes, $\pi_1(X, e)\times \pi_1(X, e)\rightarrow$ $\pi_1(X, e)$ is continuous if $\pi_1(X, e)$ is endowed with the quotient topology of the topology of compact convergence on $\Lambda_e(X)$. It is in fact the composite of the continuous isomorphism $\pi_1(X, e)\times \pi_1(X, e)\rightarrow \pi_1(X\times X,(e, e))$ (III, p. 298, Remark 3) and the continuous mapping $m_*:\pi_1(X\times X,(e, e))\rightarrow$ $\pi_1(X, e)$ (III, p. 294, Remark 1).