# T2 Space or Hausdorff Space

A Hausdorff space is a topological space in which each pair of distinct points can be separated by a 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 $U \cap V = \phi$.

Example:

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 $3,1{\text{ 3}} \in \left\{ 3 \right\},{\text{ 1}} \in \left\{ 1 \right\}{\text{ }} \Rightarrow \left\{ 3 \right\} \cap \left\{ 1 \right\} = \phi$.

Theorems
• Every metric space is a Hausdorff space.
• Every ${T_2}$ space is a ${T_1}$ space but the converse may not be true.
• Every subspace of a ${T_2}$ space is a ${T_2}$ 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, and one contains the point and the other contains the compact subspace.
• Every compact subspace of a 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 a Hausdorff space if and only if every point $a$of $X$ is the intersection of its closed neighborhoods.
• Let $X$ be a topological space and $Y$ a Hausdorff space. Let $f$ and $g$ be a 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.