# 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.