 Seifert–Van Kampen theorem

For a suitably path-connected open cover $X=U\cup V$, the fundamental group of $X$ is the pushout of the homomorphisms from ${\pi }_1(U\cap V)$ to ${\pi }_1(U)$ and ${\pi }_1(V)$.

The Seifert–Van Kampen theorem is a fundamental result in algebraic topology that provides a method for computing the fundamental group of a topological space that can be decomposed into simpler pieces. Specifically, it relates the fundamental group of a space to the fundamental groups of its subspaces when certain conditions are satisfied.