# Prove that the n-dimensional real projective space RP^n is compact

• March 26th 2008, 10:29 AM
Jonas
Prove that the n-dimensional real projective space RP^n is compact
Prove that the n-dimensional real projective space $\mathbb{RP}^n$ is compact.

I wouldn't know where to start.

Further, I have another question.

I need to prove that R with its cofinite topology is compact and sequentially compact. Compact seems easy, I did it like this:

Let (U_n) be an open cover for R, no select one (non-empty) of these open sets, say U_k0. Now, R\U_k0 is obviously finite so we can write it like {a_1,...,a_n}. Now, every one of these elements is in an open set. So, U_k0 together with these sets is an open cover.

Is this correct?
• March 26th 2008, 01:36 PM
iknowone
The real projective space is compact becuase it is the quotient of a compact space (the n dimensional unit sphere).