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 $T_\alpha = \struct {S_\alpha, \tau_\alpha}$ and $T_\beta = \struct {S_\beta, \tau_\beta}$ 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 $T_\alpha = \struct {S_\alpha, \tau_\alpha}$ and $T_\beta = \struct {S_\beta, \tau_\beta}$ 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.
Hausdorff Condition is Preserved under Homeomorphism
Let $T_A = \struct {S_A, \tau_A}, T_B = \struct {S_B, \tau_B}$ 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 $T_B$.