Separation Properties Not Preserved by Expansion

Theorem
These separation properties are not generally preserved under expansion:


 * $T_3$ Space


 * Regular Space


 * $T_4$ Space


 * Regular Space


 * $T_5$ Space


 * Normal Space


 * Completely Regular Space

Proof
Let $\left({\mathcal R, \tau_1}\right)$ be the set of real numbers under the Euclidean topology.

Let $\left({\mathcal R, \tau_2}\right)$ be the indiscrete rational extension of $\left({\mathcal R, \tau_1}\right)$.

From Metric Space fulfils all Separation Axioms, $\left({\mathcal R, \tau_1}\right)$ is:


 * $T_3$ Space


 * Regular Space


 * $T_4$ Space


 * Regular Space


 * $T_5$ Space


 * Normal Space


 * Completely Regular Space

But we have:
 * Indiscrete Rational Extension of Real Number Line is not $T_3$ Space
 * Indiscrete Rational Extension of Real Number Line is not $T_4$ Space
 * Indiscrete Rational Extension of Real Number Line is not $T_5$ Space

By definition, $\left({\mathcal R, \tau_2}\right)$ is an expansion of $\left({\mathcal R, \tau_1}\right)$.

Hence the result.