I saw this in Basic Topology by M.A.Armstrong. It gives three descriptions of real n-dimensional projective space $P^n$. Two of them are:

(a) Begin with the unit sphere $S^n$ in $E^{n+1}$ and identify its antipodal points.

(c) Begin with the unit ball $B^n$ and identify antipodal points of its boundary sphere.

I find it hard to imagine why these two descriptions lead to the same space.

Can you please help? Thank you.

  • 1
    $\begingroup$ (a) is the same as the upper hemisphere, where you identify antipodal points on the boundary $S^{n-1}$ - and that in turn is the same as (b) $\endgroup$ – user8268 Mar 29 '11 at 14:32
  • 1
    $\begingroup$ I would say (c) gives you n-1 dimensional projective space, while (a) gives you n dimensional. $\endgroup$ – GEdgar Mar 29 '11 at 14:42
  • 2
    $\begingroup$ @GEdgar , that's not true. In (c), only the antipodal points of the boundary sphere are identified. $\endgroup$ – Roun Mar 29 '11 at 14:59

Start with (a). Given $S^n$, first think about all points not on the equator (here, if $S^n = \{(x_1,...,x_{n+1})|$ $x_1^2 + ... + x_{n+1}^2 = 1\}$, then the equator is all the points with, say, $x_{n+1} = 0$).

When we identify these particular points, every point has a unique representative in the open "northern" (i.e., $x_{n+1} > 0) hemisphere. We still need to make identifications on the equatorial boundary of the closed northern hemisphere.

Thus, we can obtain $\mathbb{R}P^n$ by taking just the northern hemisphere of a sphere and identitifying some more points on the equator. But the northern hemisphere is a (closed) n-ball, and the equator is the boundary of the n-ball. Finishing up the identificatin on $S^n$ is simply a matter of identifying antipodal points on the equator, but the equator is the boundary of the n-ball, so the two constructions give the same space.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.