A Hausdorff space or T2-space is a topological space in which each pair of distinct points can be separated by disjoint open set.
            In other words, a topological space x is said to be a T2-space or Hausdorff space if for any , there exist open sets  and  such that  and .

Example:
            Let  be a non-empty set with topology  (all the subsets of X, powers set or discrete topology). Hence
            For
            For
            For  and  is a T2-space.
            For
            For
            For

Theorems:

• Every metric space is Hausdorff space.
• Every T2-space is a T1-space but the converse may not be true.
• Every subspace of T2-space is T2-space.
• In a Hausdorff space, any point and a disjoint compact subspace can be separated by open sets, in the sense that there exist disjoint open sets one contains the point and the other contains the compact subspace.
• Every compact subspace of Hausdorff space is closed.
• A one-to-one continuous mapping of a compact onto a Hausdorff space is homeomorphism.
• Let ,  be continuous functions from a space X to a Hausdorff space Y and suppose that  for all x in a dense subset D of X. Then  for all x in X.
• A space X is Hausdorff space if and only if every point a of X is the intersection of its closed neighbourhoods.
• Let X be a topological space and Y a Hausdorff space. Let  and  be continuous function from X to Y, then the set  is a closed set.