The first part of this exercise is showing that for any circle $C$ embedded in the Klein bottle $K$, $K-C$ is locally compact. This is not really hard, since $K$ is embedded in $\mathbb{R}^4$ and therefore Hausdorff. A circle is closed in $K$ and therefore $K-C$ is open. Any open in a Hausdorff space is locally compact and there we go.
However, for the second part I have to describe a circle $C$ such that $(K-C)^+$, the one point compactification of $K-C$ is homeomorphic to $\mathbb{P}^2$. Well, using your imagination in the fourth dimension is not the easiest to do :) so I hope someone comes up with an idea!
Thanks in advance!