A topological space X is said to be a T0-space 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 T0-space if for any , there exist an open set  such that  but .

Example:
            Let  with topology  defined on X, then  is a T0-space, 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 T0-space.
• Let X is a discrete topological space with at least two points, then X is not a T0-space.
• The real line  with usual topology is T0-space.
• Every sub space of T0-space is T0-space.
• A topological space X is T0-space if and only if, for any .
• Every two point co-finite topological space is a T0-space.
• Every two point co-countable topological space is a T0-space.
• If each singleton subset of a two point topological space is closed, then it is T0-space.
• If each finite subset of a two point topological space is closed, then it is a T0-space.
• Any homeomorphic image of a T0-space is a T0-space.
• A pseudo metric space is a metric space if and only if it is a T0-space.