I am not sure how to aptly phrase my question, but if we have a surface and we self intersect a part of it, will it still be homeomorphic to our original surface?
I ask because in obtaining a Klein bottle, we allow such a self intersection of a cylinder as shown in the image above. One could say that this is simply a process of creating a Klein bottle and that the cylinder is no longer homeomorphic to what we have in the image. But I am also trying to show that a projective plane with one hole is homeomorphic to a Klein bottle. A projective plane with one hole can be thought of as a disc $D^2$, identifying antipodal points on $S^1 \subset D^2$, with a hole in it. This can be thought of as a cylinder as well by considering the boundary of the disc as one hole and the hole as well another hole.
If after self intersection our surface is homeomorphic then this makes sense to me. But why is this a homeomorphism? It seems as if were identifying random sections of our surface with each other, which doesn't seem to be homeomorphic.