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 $X$ be a set.

Let $\left({X, \vartheta_1}\right)$ and $\left({X, \vartheta_2}\right)$ be topological spaces based on $X$ such that $\vartheta_2$ is an expansion of $\vartheta_1$.

That is, let $\vartheta_1$ and $\vartheta_2$ be topologies on $X$ such that $\vartheta_1 \subseteq \vartheta_2$.

Let $I_X: \left({X, \vartheta_1}\right) \to \left({X, \vartheta_2}\right)$ be the identity mapping from $\left({X, \vartheta_1}\right)$ to $\left({X, \vartheta_2}\right)$.

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

We also have that Identity Mapping is a 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.