ABSTRACT
In the previous chapter we laid the groundwork for further expansion of the homotopy theory. The Seifert-van Kampen theorem is a major step forward; it provides a way to express the fundamental group of a space in terms of the fundamental groups of open subspaces forming a cover. We will give a proof of one of the generalizations of the original statement of the theorem, then illustrate the power of the theorem through many examples.
In order to clearly see the basic idea of the Seifert-van Kampen theorem (which we will refer to as the SvK theorem), we first state it in its simplest version. It is the version that we will use most often in our applications.
