If K is the field of real numbers, and if S contains one equation x2+y2 = 1, S′ is the unit circle. If S also includes the line x = y, S′ contains only two points, one in the first quadrant and one in the third. Note that the "equations" of S are usually expressed as functions, such as x2+y2-1, and S′ defines the points that satisfy these functions, the zeros of the functions in S.
Often K is algebraically closed, and some of the theorems rely on this. Other theorems are valid for all fields.
Reversing this process, a region R in Kn has a set of functions, taken from a universal set of functions, that vanish on R. This collection of functions is denoted R′. At a minimum it includes 0.
Unless otherwise stated, the equations in S come from the polynomials in x1 through xn with coefficients in K (or perhaps a subring of K). This is a ring and a K vector space.
Note that R′ forms an ideal inside the ring of functions. Two functions that vanish on the unit circle can be added to give another function that vanishes on the unit circle, and such a function can be multiplied by any other function. If S is x2+y2-1, and R = S′ is the unit circle, R′, or S′′, includes many more functions, such as (x2+y2-1)×(x3+4xy2-17).
The closure of S, denoted S′′, is closed. This follows from the above, since S′′ is R′ for some R. Similarly, the closure of R is closed. Put this all together and S, or R, is closed iff it equals its closure.
Since R′ is an ideal, S is not closed unless it is an ideal. The closure of S includes the ideal generated by the functions of S. But an ideal need not be closed. Let x3 generate S in K[x]. Only 0 vanishes on x3, so S′ = 0. However anything generated by x vanishes at 0, so that x generates S′′, which properly contains S.
Since K adjoin finitely many indeterminants is noetherian, an ideal can be represented by a finite set of generators. A point p in Kn that satisfies every function in the ideal certainly satisfies the generators. Conversely, if p satisfies each generator it satisfies the entire ideal. Every algebraic set S′ can be represented using a finite collection of functions, assuming the functions are polynomials.