Let \(X = U_{1} \union U_{2}\), where \(U_1\), \(U_2\) and \(U_1\intersect U_2\) are open, path-connected and nonempty, and let \(x_0 \in U_1\intersect U_2\). Let \(\iota_k\colon \pi_1(U_1\intersect U_2, x_0)\to\pi_1(U_k, x_0)\), \(k = 1, 2\), be induced by the inclusions. Then the inclusions \(U_1, U_2\injects X\) induce an isomorphism \[ \pi_{1}(U_1, x_0) \ast_{\pi_{1}(U_1 \intersect U_2, x_0)} \pi_{1}(U_2, x_0) \xrightarrow{\ \sim\ } \pi_{1}(X, x_0) \] onto \(\pi_1(X, x_0)\) from the free product with amalgamation, which is the pushout of \(\pi_1(U_1, x_0)\xleftarrow{\iota_1}\pi_1(U_1\intersect U_2, x_0)\xrightarrow{\iota_2}\pi_1(U_2, x_0)\) in the category of groups [@Hat02].
Let \(X = U_1\union U_2\), where \(U_1\), \(U_2\) and \(U_1\intersect U_2\) are open, path-connected and nonempty, let \(x_0\in U_1\intersect U_2\), and let \(\iota_k\colon \pi_1(U_1\intersect U_2, x_0)\to\pi_1(U_k, x_0)\), \(k = 1, 2\), be induced by the inclusions. Suppose \[ \pi_{1}(U_1, x_0) = \left\langle u_{1}, \ldots, u_{k} \suchthat \alpha_{1}, \ldots, \alpha_{l}\right\rangle, \qquad \pi_{1}(U_2, x_0) = \left\langle v_{1}, \ldots, v_{m} \suchthat \beta_{1}, \ldots, \beta_{n}\right\rangle, \] and that \(w_1, \ldots, w_p\) generate \(\pi_1(U_1\intersect U_2, x_0)\). Then \[ \pi_{1}(X, x_0) \cong \left\langle u_{1}, \ldots, u_{k}, v_{1}, \ldots, v_{m} \suchthat \alpha_{1}, \ldots, \alpha_{l},\ \beta_{1}, \ldots, \beta_{n},\ \iota_1(w_{1}) \iota_2(w_{1})^{-1}, \ldots, \iota_1(w_{p}) \iota_2(w_{p})^{-1} \right\rangle \] [@Hat02].
The hypothesis that \(U_1 \intersect U_2\) is path-connected is necessary: for \(S^1\) covered by two open arcs, the intersection has two components, both arcs are simply connected, and \(\pi_1(S^1)\cong\ZZ\).
