There is a projection map from $R^2 \setminus (0,0)$ down to this doubled line that simply forgets the y-coordinate except at x = 0.  At x = 0 it projects y > 0 to the top origin and y < 0 to the bottom origin.

Using the open cover of the doubled line by two copies of R, one can show that this projection map is a fiber bundle with fiber R.  Thus the map from R<sup>2</sup> minus the origin to the doubled line is a weak homotopy equivalence, and so the homotopy groups of the doubled line coincide with those of S<sup>1</sup>.  The fundamental group is &#8484;.

Some entertaining generalizations of this include the fact that any finite CW-complex accepts a weak homotopy equivalence to a space with only finitely many points.