Orientation double cover of the Klein bottle

Forthe transformations with even $c$ form an index-two translation subgroup. Its quotient is a torus, giving the orientation double cover $T^2\to K$.

Let $Y$ be the mapping cylinder of the orientation double cover $T^2\to K$ with its free end omitted. If a Hausdorff space contains $Y$ as an open subset, van Kampen extends the orientation quotient of ${\pi }_1(K)$ to a surjection of the ambient fundamental group onto $C_2$. The ambient space is therefore not simply connected.