Wedge of two circles:

Let us consider the space $S^1\vee S^1$ given by the union of two circles of radius one in the plane touching each other externally at the origin. We take $U$ and $V$ to be the open sets obtained by deleting one of the points of each lobe (not the common point!). Then the circle is a deformation retract of both $U$ and $V$ and $U\cap V$ deformation retracts to the origin. Thus

$$\pi_1(S^1\vee S^1) = \mathbb{Z}*\mathbb{Z}. \eqno(26.3)$$

The last clause in theorem (26.1) also provides the generators of the fundamental group. Assuming the circles to centered at $\pm 1$, the generators are given by the homotopy classes of the loops

$$\pm 1 + \exp(2\pi i t) \eqno(26.4)$$

The generalization to a wedge of $n$ circles is left as an exercise.