 Solution

Let $X=Y\cup Z$, where $Y$, $Z$, and $Y\cap Z$ are path-connected subcomplexes containing the basepoint; small open neighbourhoods may be used to reduce to the usual open-cover statement. The Seifert-van Kampen theorem says thatMore explicitly, this is the quotient of the free product ${\pi }_1(Y)\ast {\pi }_1(Z)$ by the normal closure ofEquivalently, ${\pi }_1(X)$ is the pushout of the two inclusion-induced homomorphisms.