The following is the definition of Zariski topology
I am reading a theorem of Lie algebra.In its proof,he says U and R are Zariski open subsets: 
I have problems in geting the polynomials and showing that they aren't the common zeros of these polynomials. Can anyone help me? Thank you.
The conditions on a linear operator to be nilpotent, respectively singular, are polynomial in whatever coordinates you introduce on your space(s). Indeed, if $T$ is a linear operator on a space of dimension $N$, then $T$ is nilpotent if and only if $T^N = 0$ - and $T^N$ entries (matrix elements if you will) are polynomial in the matrix elements of $T$; and $T$ is singular if and only if $\det T = 0$, and $\det$ is also polynomial in $T$.