Separation Properties Preserved by Expansion

Theorem
These separation properties are preserved under expansion:


 * $T_0$ (Kolmogorov) Space


 * $T_1$ (Fréchet) Space


 * $T_2$ (Hausdorff) Space


 * $T_{2 \frac 1 2}$ (Completely Hausdorff) Space

Proof
Let $S$ be a set.

Let $\struct {S, \tau_1}$ and $\struct {S, \tau_2}$ be topological spaces based on $S$ such that $\tau_2$ is an expansion of $\tau_1$.

That is, let $\tau_1$ and $\tau_2$ be topologies on $S$ such that $\tau_1 \subseteq \tau_2$.

Let $I_S: \struct {S, \tau_1} \to \struct {S, \tau_2}$ be the identity mapping from $\struct {S, \tau_1}$ to $\struct {S, \tau_2}$.

From Identity Mapping to Expansion is Closed, we have that $I_S$ is closed.

We also have Identity Mapping is Bijection.

So we can directly apply:
 * $T_0$ (Kolmogorov) Space is Preserved under Closed Bijection


 * $T_1$ (Fréchet) Space is Preserved under Closed Bijection


 * $T_2$ (Hausdorff) Space is Preserved under Closed Bijection


 * $T_{2 \frac 1 2}$ (Completely Hausdorff) Space is Preserved under Closed Bijection

and hence the result.