i think you are misunderstanding what he is doing. a given line L is not an open set in RP2. imagine it this way: suppose we have an open disk in R2. now add a third dimension, and "project" the disk into a cone (through the origin). open sets in RP2 will look like unions of these cones. in other words, open sets are "open bundles of lines".
it might be easier to see what is happening in 2 dimensions. a "basic open set" in the real line is an open interval. now, draw a line parallel to the x-axis in R2, and consider the set of lines in R2 that pass through the open interval translated to that parallel line. what you get is essentially "two angle sectors", located on opposite sides of the unit circle (or any other circle centered at the origin, for that matter).
it would be a mistake to think that all open sets in RP2 are "cone-shaped". it is better to think of them as "projections" of open sets in R2 through the origin (the projections look "flipped around" on the lower half-space determined by z = 0). something "impossible" happens at the origin (the shape of the projection isn't maintained), so it's best to avoid it.