Zariski closure

Zariski closure is a concept in algebraic geometry that describes how a set of points relates to the smallest algebraic shape (like a curve or surface) containing them. Think of it as finding the most fundamental geometric object defined by polynomial equations that includes all given points. This "closure" captures all points that satisfy these defining relations, even if they weren't originally in the set. It helps mathematicians understand how points are connected through underlying algebraic structures and provides a way to study the properties of geometric objects via their polynomial equations.