In D-Branes by Clifford V. Johnson page 56
the projective plane $\mathbb{R P}^2$, which is a disk with opposite points identified (see figure 2.13).
But the real projective plane was explained to be
A projective plane can be thought of as an ordinary plane equipped with additional "points at infinity" where parallel lines intersect. Thus any two distinct lines in a projective plane intersect at exactly one point.
Then how was of the opposite points identified? Were they identified as a single point $(-\infty,\infty)$ or were they identified through $(x,y)\cong(-x,-y)$?
Shrinking the identified hole down, we recover the fact that $\mathbb{R P}^2$ may be thought of as a sphere with a crosscap inserted, where the crosscap is the result of shrinking the identified hole. Actually, a M¨obius strip can be thought ofas a disc with a crosscap inserted, and a Klein bottle is a sphere with two crosscaps.
What was the identified hole hole? Was it the boundary of the disk? Also, according to the website ,
In mathematics, a cross-cap is a two-dimensional surface in 3-space that is one-sided and the continuous deformation of a Klein bottle that intersects itself in an interval. In the domain, the inverse image of this interval is a longer interval that the mapping into 3-space "folds in half". At the point where the longer interval is folded in half in the image, the nearby configuration is that of the Whitney umbrella.
However, the disk was oriented. How to insert the unoriented surface(one sided) on an oriented surface(two sided)? Was this "crosscap" inserted at the boundary or in the bulk of the plane?
What is the crosscap of the projective plane $\mathbb{R P}^2$?