In Kosniowski's book "A First Course in Algebraic Topology", he used Van Kampen's theorem to calculate the fundamental group of given pictures. He defined the first open set, for example for $X_1$ is $U_{1,1}=X_1-b$. But I think $U_{1,1}$ is not path-connected, so Van Kampen's theorem cannot be used.
If I am wrong, can someone explain why $U_1$ is path-connected? Maybe he means that in $X_1-b$ we also remove the region bounded by $b$?