Seifert–van Kampen theorem

A full proof requires careful combinatorial bookkeeping; here is the structural outline. Step 1 — Generators. Any loop $\gamma$ in $X$ based at $x_0$ can be subdivided into finitely many sub-paths, each lying entirely in $U$ or entirely in $V$ (using a Lebesgue number argument on the open cover $\{\gamma^{-1}(U), \gamma^{-1}(V)\}$ of the compact interval $[0,1]$), so $[\gamma]$ can be written as a product of loops each representing an element of $\pi_1(U,x_0)$ or $\pi_1(V,x_0)$ (after connecting sub-path endpoints back to $x_0$ through chosen paths in $U \cap V$, which is path-connected). This shows $\pi_1(X,x_0)$ is generated by the images of $\pi_1(U,x_0)$ and $\pi_1(V,x_0)$.

Step 2 — Relations. Any loop $\delta$ lying in $U \cap V$ represents, a priori, two possibly-different elements: $i_U(\delta) \in \pi_1(U,x_0)$ (viewing $\delta$ as a loop in $U$) and $i_V(\delta) \in \pi_1(V,x_0)$ (viewing it as a loop in $V$). Since $\delta$ is literally the same loop in $X$, its images under $\pi_1(U) \to \pi_1(X)$ and $\pi_1(V) \to \pi_1(X)$ must agree; this forces exactly the amalgamation relations $i_U(\delta) = i_V(\delta)$ for every $[\delta] \in \pi_1(U \cap V, x_0)$.

Step 3 — No further relations. A more delicate argument (subdividing homotopies $H: [0,1]\times[0,1] \to X$ the same way, using compactness of the square to get a finite grid where each cell lies in $U$ or $V$) shows that these amalgamation relations are the only relations needed: any two words in the generators representing the same element of $\pi_1(X,x_0)$ can be related by a finite sequence of moves each justified by an amalgamation relation. This identifies $\pi_1(X,x_0)$ exactly with the amalgamated free product.