A topological space X is said to be a T0space if for any pair of distinct points of X, there exist at least one open set which contains one of them but not the other.
In other words, a topological space X is said to be a T0space if for any , there exist an open set such that but .
Example:
Let with topology defined on X, then is a T0space, because

for a and b, there exist an open set such that and

for a and c, there exist an open set such that and

for b and c, there exist an open set such that and
Theorems:

Let X is an indiscrete topological space with at least two points, then X is not a T0space.

Let X is a discrete topological space with at least two points, then X is not a T0space.

The real line with usual topology is T0space.

Every sub space of T0space is T0space.

A topological space X is T0space if and only if, for any .

Every two point cofinite topological space is a T0space.

Every two point cocountable topological space is a T0space.

If each singleton subset of a two point topological space is closed, then it is T0space.

If each finite subset of a two point topological space is closed, then it is a T0space.

Any homeomorphic image of a T0space is a T0space.

A pseudo metric space is a metric space if and only if it is a T0space.
