Zariski topology on affine space

The Zariski topology on affine space is a way of defining "closeness" or "nearness" based on algebraic equations. In this topology, the basic closed sets are the common solutions to a collection of polynomial equations, called algebraic varieties. Unlike usual notions of closeness, these sets can be very large or sparse and may not resemble familiar geometric shapes. Open sets are complements of these solution sets. This topology is fundamental in algebraic geometry because it captures the geometric structure dictated by polynomial equations, emphasizing the relationship between algebraic formulas and geometric objects.