A topological space X is said to be a T1-space if for any pair of distinct points of X, there exist two open sets which contain one but not the other.
            In other words, a topological space X is said to be a T1-space if for any  there exist open sets  and  such that  and .

Example:
            Let  with topology  defined on X is not a T1-space because for , we have open sets  and X such that . This shows that we cannot find an open set which contains c but not a, so  is not a T1-space. But we have already showed that  is a T0-space. This shows that a T0-space may not be a T1-space. But the converse is always true.

Example:
            The real line  with usual topology is T1-space.
            To prove this, we suppose that . Further assume that . Since the usual topology on  consists of open intervals, so we have open sets  and , such that  and . This shows that the real line  with usual topology is a T1-space.

Theorems:

• Every T1-space is a T0-space
• An indiscrete topological space with at least two points is not a T1-space.
• The discrete topological space with at least two points is a T1-space.
• Every two point co-finite topological space is a T1-space.
• Every two point co-countable topological space is a T1-space.
• Every subspace of T1-space is T1-space.
• A topological space is a T1-space if and only if its each finite subsets is a closed set.
• Following statements about a topological space X are equivalent. (1) X ia a T1-space. (2) Each singleton subset of X is closed. (3) Each subset A of X is the intersection of its open supersets.
• Any homeomorphic image of a T1-space is a T1-space.
• If x is a limit point of a set A in a T1-space X, then every open set containing an infinite number of distinct points of A.
• A finite set has no limit points in a T1-space.