T4 Property is not Hereditary

Theorem

Let $T = \struct {S, \tau}$ be a topological space which is a $T_4$ space.

Let $T_H = \struct {H, \tau_H}$, where $\O \subset H \subseteq S$, be a subspace of $T$.


Then it does not necessarily follow that $T_H$ is a $T_4$ space.


Proof

Let $T$ be the Tychonoff plank.

Let $T'$ be the deleted Tychonoff plank.

By definition, $T'$ is a subspace of $T$.


From Tychonoff Plank is Normal, $T$ is a normal space.

From Deleted Tychonoff Plank is Not Normal, $T'$ is not a normal space.

Thus it is seen that the property of being a normal space is not inherited by a subspace.

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
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