The boundary curves mentioned in (3) are called top cycles, and they play a key role in understanding tight immersions. Note that a top cycle is an embedded, planar, convex curve, and that every top cycle has an orientable neighborhood .
If a tight immersion of a connected surface has no top cycles, then it is necessarily a sphere, since in this case the image of the M+ region is all of the convex envelope (and there is no M- region). So a tight immersion of the projective plane must have at least one top cycle.
There is only one class of embedded curves on the projective plane that do not bound regions; but curves in this class have non-orientable neighborhoods (the neighborhood is a Möbius band, as in the diagram below), and so they can not be top cycles.
This is a contradiction, so there can be no tight immersion of the real projective plane.
7/21/94 email@example.com --
The Geometry Center