AlgTop7b: The Klein bottle and projective plane (cont.)

[Toxie207]

1:50 But isn't the loop the union of all the lines intersecting at that point? Maybe I misunderstand what a 'loop' is?

[njwildberger]

Hi@Toxie207 It is a subtle issue. Our space M in question is the space of all lines in the plane. It is not the plane itself. Each line passing through the given point represents a distinct point in M. So as we rotate those lines about the fixed point, we get a loop of distinct points in M. The trick is in realizing that the space of all lines M is itself a two dimensional surface, and we are trying to figure out what it looks like.