Define an equivalence relation on by writing if and only if . The quotient space is called projective -space. (This is one of the ways that we defined the projective plane .) The canonical projection is just . Define , , by setting
1) is open in .
2) covers .
3) There is a homeomorphism .
4) is compact, connected, and Hausdorff, hence is an -manifold.
1) is open in , so is open in .
2) Let . Then since , so . Hence .
3) Consider . Define for . If , then . Then is well-defined.
where . Both and are continuous, so is a homeomorphism.
4) Since is compact and connected, so is . is a CW-complex with one cell in each dimension, i.e.\ . Since CW-complexes are Hausdorff, so is .