Separation Properties Not Preserved by Expansion

Theorem
These separation properties are not generally preserved under expansion:


 * $T_3$ Space


 * Regular Space


 * $T_4$ Space


 * Completely Regular Space


 * $T_5$ Space


 * Normal Space


 * Completely Normal Space

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

Let $\struct {\R, \tau_2}$ be the indiscrete rational extension of $\struct {\R, \tau_1}$.

From Metric Space fulfils all Separation Axioms, $\struct {\R, \tau_1}$ is:


 * $T_3$ Space


 * Regular Space


 * $T_4$ Space


 * Completely Regular Space


 * $T_5$ Space


 * Normal Space


 * Completely Normal 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, $\struct {\R, \tau_2}$ is an expansion of $\struct {\R, \tau_1}$.

Hence the result.