The real projective space is compact becuase it is the quotient of a compact space (the n dimensional unit sphere).
Prove that the n-dimensional real projective space 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?