The Zariski topology is a topology that is well-suited for the study of polynomial equations in algebraic geometry, since a Zariski topology has many fewer open sets than in the usual metric topology. In fact, the only closed sets are the algebraic sets, which are the zeros of polynomials.
For example, in 
,
the only nontrivial closed sets are finite sets of points. In 
, there are also the zeros of polynomials such as lines 
and cusps 
.
On 
for 
,
the Zariski topology is not a T2-space.
Any two nonempty open sets intersect.
Every nonempty open set is dense
both in the Zariski topology and in the usual metric
topology. These statements hold because affine space
is an irreducible variety and do not hold
for arbitrary algebraic sets.
Because there are fewer open sets than in the usual topology, it is more difficult for a function
to be continuous in Zariski topology. For example,
a continuous function 
must be a constant
function. Conversely, when the range has the Zariski
topology, it is easier for a function to be continuous.
In particular, the polynomials are continuous
functions 
.
In general, the Zariski topology of a commutative ring 
with multiplicative identity is a topology
on the set of prime ideals,
known as the ring spectrum. Its closed
sets are 
,
where 
is any ideal in 
and 
is the set of prime
ideals containing 
.
