# Properties of Hausdorff Space

## Theorem

### Subspace of Hausdorff Space is Hausdorff

Let $T = \struct {S, \tau}$ be a topological space which is a $T_2$ (Hausdorff) space.

Let $T_H = \struct {H, \tau_H}$, where $\O \subset H \subseteq S$, be a subspace of $T$.

Then $T_H$ is a $T_2$ (Hausdorff) space.

That is, the property of being a $T_2$ (Hausdorff) space is hereditary.

### Product of Hausdorff Factor Spaces is Hausdorff

Let $T_\alpha = \left({S_\alpha, \tau_\alpha}\right)$ and $T_\beta = \left({S_\beta, \tau_\beta}\right)$ be topological spaces.

Let $T = T_\alpha \times T_\beta$ be the product space of $T_\alpha$ and $T_\beta$

Let $T_\alpha$ and $T_\beta$ both be $T_2$ (Hausdorff) spaces.

Then $T$ is also a $T_2$ (Hausdorff) space.

### Domain of Continuous Injection to Hausdorff Space is Hausdorff

Let $T_\alpha = \left({S_\alpha, \tau_\alpha}\right)$ and $T_\beta = \left({S_\beta, \tau_\beta}\right)$ be topological spaces.

Let $f: S_\alpha \to S_\beta$ be a continuous mapping which is an injection.

If $T_\beta$ is a $T_2$ (Hausdorff) space, then $T_\alpha$ is also a $T_2$ (Hausdorff) space.

### $T_2$ (Hausdorff) Space is Preserved under Homeomorphism

Let $T_A = \left({S_A, \tau_A}\right), T_B = \left({S_B, \tau_B}\right)$ be topological spaces.

Let $\phi: T_A \to T_B$ be a homeomorphism.

If $T_A$ is a $T_2$ (Hausdorff) space, then so is $T_B$.