Klein-bottle mapping-cylinder obstruction to simple connectivity
= Klein-bottle mapping-cylinder obstruction to simple connectivity
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.