Corollary 2

Let X and Y be topological spaces and let $f: X\rightarrow$ Y be a homeomorphism. For every point $x$ of X, the homomorphism

is an isomorphism.

Let $g$ be a continuous mapping of Y into X, inverse to $f$ up to homotopy. Let $x$ be a point of X. It follows from Prop. 3 applied to the homotopic mappings Id$_X$ and $g\circ f: X\rightarrow X$ that the mapping $\pi_1(g\circ f, x)$ is an isomorphism of the group $\pi_1(X, x)$ onto the group $\pi_1(X, g\circ f(x))$. Since

the homomorphism $\pi_1(f, x)$ is injective and the homomorphism $\pi_1(g, f(x))$ is surjective. Since the mapping $g$ is also a homotopy equivalence, the homomorphism $\pi_1(g, f(x))$ is injective; it is therefore an isomorphism. Hence $\pi_1(f, x)$ is an isomorphism.