Example

Let G be a topological group, let $e$ be its identity element. For every point $g$ of G, the left and right translations, $x\mapsto gx$ and $x\mapsto xg$, are homeomorphisms of G onto itself (TG, III, p. 2) which map $e$ onto $g$. By Corollary 2, they induce isomorphisms of $\pi_1(G, e)$ onto $\pi_1(G, g)$. These isomorphisms are not necessarily equal (IV, p. 459, exerc. 1).