Separation Properties Preserved in Subspace/Corollary
Theorem
Let $T = \struct {S, \tau}$ be a topological space.
Let $T_H$ be a subspace of $T$.
If $T$ has one of the following properties then $T_H$ has the same property:
That is, the above properties are all hereditary.
Proof
A regular space is a topological space which is both a $T_0$ (Kolmogorov) space and a $T_3$ space.
Hence from $T_0$ Property is Hereditary and $T_3$ Property is Hereditary it follows that the property of being a regular space is also hereditary.
A Tychonoff (completely regular) space is a topological space which is both a $T_0$ (Kolmogorov) space and a $T_3 \frac 1 2$ space.
Hence from $T_0$ Property is Hereditary and $T_3 \frac 1 2$ Property is Hereditary it follows that the property of being a Tychonoff (completely regular) space is also hereditary.
A completely normal space is a topological space which is both a $T_1$ (Fréchet) space and a $T_5$ space.
Hence from $T_1$ Property is Hereditary and $T_5$ Property is Hereditary it follows that the property of being a completely normal space is also hereditary.
$\blacksquare$
$T_4$ Space
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