Separation Properties Preserved in Subspace/T4 Space

Theorem

Of all the separation axioms, the $T_4$ axiom differs from the others.

It does not necessarily hold that a subspace of a $T_4$ space is also a $T_4$ space, unless that subspace is closed.

This is demonstrated in the result $T_4$ Property is not Hereditary.

However, it is the case that the $T_4$ property is weakly hereditary.

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