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. Explicitly, there is a homeomorphism $$ f(x,y) = (x, y/4 - x^2/y) $$ from $R \times (0,\infty)$ to $R^2 \setminus \{y < 0\}$ with inverse $$ g(x,u) = (x,-2u + 2\sqrt{u^2 + x^2}). $$ (This may look more complicated than it really is; the function f takes the lines $y = c$ and turns them into a sequence of parabolae foliating $R^2 \setminus \{y < 0\}$.

Thus the map from R² 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 S¹. The fundamental group is ℤ.

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.

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. Explicitly, there is a homeomorphism $$ f(x,y) = (x, y - x^2/4y) $$ from $R \times (0,\infty)$ to $R^2 \setminus \{(0,y): y \leq 0\}$ with inverse $$ g(x,u) = (x,(u + \sqrt{u^2 + x^2})/2). $$ (This may look more complicated than it really is; the function f takes the lines $y = c$ and turns them into a sequence of parabolae foliating $R^2 \setminus \{(0,y): y \leq 0\}$.

Thus the map from R² 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 S¹. The fundamental group is ℤ.

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.

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 R² 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 S¹. The fundamental group is ℤ.

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.

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. Explicitly, there is a homeomorphism $$ f(x,y) = (x, y/4 - x^2/y) $$ from $R \times (0,\infty)$ to $R^2 \setminus \{y < 0\}$ with inverse $$ g(x,u) = (x,-2u + 2\sqrt{u^2 + x^2}). $$ (This may look more complicated than it really is; the function f takes the lines $y = c$ and turns them into a sequence of parabolae foliating $R^2 \setminus \{y < 0\}$.

Thus the map from R² 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 S¹. The fundamental group is ℤ.

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.