I am trying to follow the proof in Loring Tu's book (An introduction to smooth manifolds, 2nd edition, p.79) to show that the real projective space is Hausdorff. A snippet of the proof is shown below. 
I am confused about the part where we show that $R$ is a closed subset. I understand that the rank of $x$ and $y$ concatinated being at most 1 is equivalent to the vanishing of all $2\times 2$ minors of $[x \hspace{0.5em} y]$. My 2 questions are:
1) How is $R$ the zero set of finitely many polynomials? Showing the rank is at most 1 only involves $2 \times 2$ minors, i.e. only 2 rows of $R$ are utilized.
2) If it is the zero set, why is it closed? Does the "finitely" (in "finitely many polynomials) play a role here?
Hoping someone can help clarify things. Thanks!
That's a rather baroque proof!
If we write $x=(x_0,\ldots,x_n)$ and $y=(y_0,\ldots,y_n)$ then the finitely many equations are the $x_iy_j-x_jy_i=0$ or $0\le i< j\le n$.
By continuity, the subset defined by one equation $x_iy_j-x_jy_i=0$ is a closed subset. The subset defined by all of them is the intersection of these closed subset, and is also closed. (Finiteness is not essential here.)