Remark 2

Let B be a topological space. Let $b$ and $b'$ be points of B which belong to the same path-connected component and let $\gamma$ be an element of $\varpi_{b,b'}(B)$. It follows from the definition of an admissible subgroup that, for every admissible subgroup H of $\pi_1(B, b')$, the subgroup $\gamma H\gamma^{-1}$ of $\pi_1(B, b)$ is admissible. Consequently, the isomorphism $u_{\gamma}:\delta \mapsto \gamma \delta \gamma^{-1}$ of $\pi_1(B, b')$ onto $\pi_1(B, b)$ (cf. III, p. 292) is a homeomorphism when these groups are endowed with the admissible topologies.