For example, in ,
the only nontrivial closed sets are finite collections of points. In , there are also the zeros of polynomials such as lines and cusps .

The Zariski topology is not a T2-space. In fact, any two open sets must intersect, and cannot be disjoint.
Also, the open sets are dense, in the Zariski topology
as well as in the usual metric topology.

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 ring 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 .