Homotopy Invariance of the Fundamental Groupoid

Let \(X\) and \(Y\) be topological spaces.

Theorem 1 (Induced Functors) A continuous map \(f : X \to Y\) induces a well-defined functor: \[\Pi_1(f) : \Pi_1(X) \to \Pi_1(Y)\]

Proof. For every path class \([\gamma] \in \Pi_1(X)(x_0, x_1)\), define \(\Pi_1(f)([\gamma]) = [f \circ \gamma]\)…