The Klein bottle $\displaystyle K^2$ is a square where the opposite vertical edges are identified in the opposite direction and the horizontal edges are identified in the same direction.

Consider the space $\displaystyle \mathbb{R}P^2 \# \mathbb{R}P^2$ resulting from an annulus by identifying antipodal points on the outer circle, and also identifying antipodal points in the inner circle. Show that $\displaystyle K^2 \cong \mathbb{R}P^2 \# \mathbb{R}P^2$, i.e. $\displaystyle K^2$ is homeomorphic to $\displaystyle \mathbb{R}P^2 \# \mathbb{R}P^2.$