Zariski topology

The Zariski topology is a topological space with closed sets which are the vanishing loci of algebra ideals of for an algebraically closed field .

With the Zariski topology, is called the affine n-space over , and is denoted .

Proof this is a topology

The set of vanishing ideals must satisfy the axioms of a topological space. Let

1. and .

2. Given ideals ,

3. Given ideals we have

Why this topology

Given a continuous function then is continuous. Therefore, the Zariski topology is restricting the possible closed sets such that only polynomials are continuous.

This means that since given the Euclidean topology polynomials are continuous, that the Zariski topology has far fewer open (or closed) sets and is much more restrictive.

Zariski topology on prime spectrum.

For a ring and a subset , let

Note that if is the ideal generated by then

Thus, the sets

form the closed sets of a topology on .

Proof

First, and .

Next, if is a family of ideals in , then we claim

Proof: If , then

therefore, we have

This gives

Say , that is, . Then for all , so

Lastly, if are ideals, then we claim

Proof: Say , that is . Say . Then there exists . For any , we have . Since is prime, then , that is . So , giving

Say , that is . Let and . Let , then

However, , so that . This gives

Induced topology on \text{MSpec }R.

Say . Then the subspace topology on is the “normal” Zariski topology described above.

Open sets of Zariski topology

For , the sets

are an open topological basis for the Zariski topology.

Quasi-compact

is “quasi-compact”. That is, each open set is compact.