The above video describes the real projective plane ().

The projective space can be defined as the quotient space of by the equivalence relation for .

Notation: For , we write for the corresponding point in . Let be the maps defined by and .

How do we construct an explicit homotopy between and ? A common mistake is to try the “straight-homotopy”, e.g. . This is a mistake as it passes through the point [0,0,0] which is not part of the projective plane.

A better approach is to consider , defined by .

Note that if , then .