  As we known, the {\it Seifert-Van Kampen theorem} handles fundamental groups
of those topological spaces $X=U\cup V$ for open subsets $U, V\subset X$ such
that $U\cap V$ is arcwise connected. In this paper, this theorem is generalized
to such a case of maybe not arcwise-connected, i.e., there are $C_1$,
$C_2$,$..., C_m$ arcwise-connected components in $U\cap V$ for an integer
$m\geq 1$, which enables one to find fundamental groups of combinatorial spaces
by that of spaces with theirs underlying topological graphs, particularly, that
of compact manifolds by their underlying graphs of charts.

General Mathematics