T2 Space or Hausdorff Space

A Hausdorff 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 {T_2} space or Hausdorff space if for any x,y \in X,{\text{ }}x \ne y, there exist open sets U and V such that x \in U,{\text{ }}y \in V and U \cap V = \phi .


Let X = \left\{ {1,2,3} \right\} be a non-empty set with topology \tau  = P\left( X \right) (all the subsets of X, powers set or discrete topology). Hence
For 1,2{\text{   }}1 \in \left\{ 1 \right\},{\text{ }}2 \notin \left\{ 1 \right\}
For 2,3{\text{   2}} \in \left\{ 2 \right\},{\text{ 3}} \notin \left\{ 2 \right\}
For 3,1{\text{   3}} \in \left\{ 3 \right\},{\text{ 1}} \notin \left\{ 3 \right\} and \left( {X,\tau } \right) is a {T_2} space.
For 1,2{\text{   }}1 \in \left\{ 1 \right\},{\text{ }}2 \in \left\{ 2 \right\}{\text{ }} \Rightarrow \left\{ 1 \right\} \cap \left\{ 2 \right\} = \phi
For 2,3{\text{   2}} \in \left\{ 2 \right\},{\text{ 3}} \in \left\{ 3 \right\}{\text{ }} \Rightarrow \left\{ 2 \right\} \cap \left\{ 3 \right\} = \phi
For 3,1{\text{   3}} \in \left\{ 3 \right\},{\text{ 1}} \in \left\{ 1 \right\}{\text{ }} \Rightarrow \left\{ 3 \right\} \cap \left\{ 1 \right\} = \phi .

• Every metric space is Hausdorff space.
• Every {T_2} space is a {T_1} space but the converse may not be true.
• Every subspace of {T_2} space is {T_2} 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 f:X \to Y, g:X \to Y be continuous functions from a space X to a Hausdorff space Y and suppose that f\left( x \right) = g\left( x \right) for all x in a dense subset D of X. Then f\left( x \right) = g\left( x \right) for all x in X.
• A space X is Hausdorff space if and only if every point aof X is the intersection of its closed neighbourhoods.
• Let X be a topological space and Y a Hausdorff space. Let f and g be continuous function from X to Y, then the set A = \left\{ {x \in X:f\left( x \right) = g\left( x \right)} \right\} is a closed set.