The Zariski Topology

First introduced by Oscar Zariski as a way of turning zero sets of polynomials into topological spaces, the Zariski topology has played an important role in the development of modern algebraic geometry. A particularly interesting attribute of this topology is that it is almost never Hausdorff (T_2), making it one of the prototypical examples of a non-Hausdorff space. However, the Zariski topology is T_1 for one-dimensional rings, and it is always T_0. In this talk, I will define the Zariski topology in a more general setting---on the spectrum of a commutative ring with identity, which provides the starting point for the construction of affine schemes---giving extensive examples to illustrate its construction and separation properties.