Define an equivalence relation on
by writing
if and only if
. The quotient space
is called the (real) projective n-space.
Let ,
be the canonical
projection. For each i, ,
define a subset of
by
Prove the following facts, which together show that
is an n-manifold.
- is open in
.
- covers .
- There is a homeomorphism .
- is compact, connected, and Hausdorff.