My question is in relation to the accepted answer here .
I have already proven that the fundamental group of the plane minus two points is just the same as wedge of two circles and hence is $\Bbb{Z}*\Bbb{Z}$ by considering the plane as an open disc and removing two points from the interior and then using Van Kampen's Theorem.
But I do not know how to compute the fundamental group of disc minus two points . As far as I can see there should be a few cases. For example if I remove two points from the interior then I should again be able to use Van-Kampen and conclude that it is the same as that of the wedge of two circles.(Although I am not very sure as to what the open sets should be. But I guess I'll need to use the subspace topology and modify the way I proved for the open disc a little).
But what if the two points are on the boundary or one point in the interior and the other on the boundary. In these cases I am having trouble as to apply Van Kampen as I cannot figure out what open sets should I choose.
Any help is appreciated .