I am taking a course on Algebraic geometry this semester and this proof was left as an exercise. I am struck in one argument of the proof, so I request you to help me.
Statement: Let E be a subset of $K^n$. Then show that $V_K(I_K(E))= \overline{E}$. Here Symbol V means a affine algebraic set and $I_K$ is a map opposite in the direction to the map $A\to V_K(A)$ , where $A$ is an ideal in $R= K[X_1,...,X_n]$.
Attempt: I am not able to prove that $E\subseteq V_K(I_K(E))$. T took and element $ x\in E$ but I am not sure why it should always lies in $V_K(I_K(E))$ ? Range of $V_K(I_K(E))$ is $K^n$ .
Then as $V_K(I_K(E))$ is a closed set in Zariski Topology , this side of proof will be done.
Converse side I have done by taking an ideal $ b$ in R and the using the definition of $I_K$ in terms of maximal ideals.
Kindly guide!
If $f\in I(E)$, then $f(x)=0$ for all $x\in E$. So $V(I(E))$ certainly contains $E$.