Circle such that the one point compactification of the Klein bottle minus the circle is homeomorphic to $\mathbb{P}^2$ 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!
 A: You are right, using your imagination in the fourth dimension is not the easiest to do. But this is a doubtless useful skill for a topologist. Charles H. Hinton argued that it can be developed by freeing our imagination of objects of “elements of the self”, related to our vision and location.  But I read that «Hinton later introduced a system of coloured cubes by the study of which, he claimed, it was possible to learn to visualise four-dimensional space (Casting out the Self, 1904). Rumours subsequently arose that these cubes had driven more than one hopeful person insane». So I better answer your question before I’ll start to apply his methods. 
Since your asked for an idea, I followed my topological intuition and imagination and didn’t check the details. As three dimensional beings, we could consider a usual quasiembedding of the bottle in our space, but I think it is easier and rigorous to use old and steady plane model of the Klein bottle a quotient space of a square with the sides glued according to the arrows.    

We have to find a circle such that we obtain the projective plane when we collapse the circle to a point. An easy candidate is a vertical side, which will become a circle when all corners of the square will be glued together. Indeed, when we collapse the vertical sides of the square to points, we obtain a usual disk model of the projective plane. I remark that since a composition of quotient maps is a quotient map, the order of gluing should not change the topology of the final space. 
