I'm reading a book by W. Fulton about algebraic geometry, and it states a property as follows:
$I(V (I(X))) = I(X)$ for any set $X$ of points. So if $I$ is the ideal of an algebraic set, $I = I(V (I))$.
But aren't this things the same, because if $I$ is the ideal of an algebraic set, ($I=I(X)$ where $X$ is algebraic) then
$$I(X)=I(V(I(X)))$$
Which is exactly what the first part told me in general, why is it necessary to specify the case when $X$ is algebraic?
The author is most likely repeating the concept with the intent of stressing on the fact that $I(V(I))$ does not hold for all ideals, but just for some of them. It's noteworthy that we do not lose generality in assuming that $X$ is an algebraic set: for all $X\subseteq k^n$ there is some algebraic set $Y$ such that $I(X)=I(Y)$; namely, $Y=V(I(X))$ as the author has just pointed out.