In this lecture we finally prove something difficult: the Seifert-van Kampen Theorem. This theorem is extremely helpful in computing the fundamental group of many “real-life”  spaces, as you'll discover on Problem Sheet C.

Sadly however we can't recover the fact that $\pi_1(S^1) \cong \mathbb{Z}$ using this result, since the circle can't be written as the union of two open sets with connected intersection. This can be rectified by using the “groupoid” version of the Seifert-van Kampen Theorem, but this is too advanced to cover here.