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.

$\blacksquare$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $2$: Separation Axioms: Functions, Products, and Subspaces