Separation Properties Not Preserved by Expansion
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Theorem
These separation properties are not generally preserved under expansion:
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:
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