**Definitions.**

- (a)
- An ideal of a ring is called
*prime*if whenever for , then or (or both). - (b)
- A variety is called
*irreducible*if for any decomposition of into a union of two subvarieties, either or . In other words, there are no non-trivial decompositions of into smaller varieties.

Thus, there exists a one-to-one correspondence between the set of varieties and the set of prime (homogeneous if projective ) ideals in the corresponding polynomial ring.

The varieties appearing in this unique decomposition
are called the *irreducible components* of .

**Definition.** Let be a set of points in some
space. A *topology* on is a set
of designated
subsets of , called the *open* sets of , so that the
following axioms are satisfied:

- (a)
- The union of any collection of open sets is open.
- (b)
- The intersection of any finite collection of open sets
is open.
- (c)
- and are open.

The *closed* sets in are the complements of the
open sets.

We define below the so-called *Zariski topology* on
algebraic varieties. If we work over
, every variety can
be roughly viewed as a *complex manifold* (with the exception
of a proper subset of *its singular points*). Through its
embedding in, say,
, will inherit the usual *complex analytic* topology from
- a basis for the open
sets on will consist of the intersections of with any finite
balls in
.

The

**Exercise.** Show that the Zariski topology on the
projective line
is different from the
analytic topology of
.

Many statements in algebraic geometry are true for