I have to calculate the fundamental group of $$\left\{(x-1,y) | (x,y)\in \Bbb S^{1}\right\}\cup(\{0\}\times\mathbb{R})$$ via Van Kampen's Theorem, but I have no idea on how to start. What opens subspaces U, V I could use? Thanks.
My attempt: I think the fundamental group is the same as the circle, i.e., $\Bbb Z$ because the line is tangent to the circle. I´m right?
The standard approach with Van Kampen in these exercises is to let $U_0$ be a small open neighborhood of the point where the line and the circle are glued together (i.e. the origin). Then set $U$ to be the union of $U_0$ with the circle, and set $V$ to be the union of $U_0$ with the line.
Since both $V$ and $U_0=U\cap V$ are null-homotopic, we get from the theorem that $U$ and your space have isomorphic fundamental groups. And the fundamental group of $U$ is clearly the fundamental group of the circle (they are, for instance, easily seen to be homotopy equivalent, and what's more, the circle is a deformation retract of $U$).