Separation Properties Preserved in Subspace

Theorem
Let $T = \displaystyle \left({X, \tau}\right)$ 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:


 * $T_0$ (Kolmogorov) Property


 * $T_1$ (Fréchet) Property


 * $T_2$ (Hausdorff) Property


 * $T_{2 \frac 1 2}$ (Completely Hausdorff) Property


 * $T_3$ Property


 * $T_{3 \frac 1 2}$ Property


 * $T_5$ Property

That is, the above properties are all hereditary.

Corollary
If $T$ has one of the following properties then $T_H$ has the same property:


 * Regular Property


 * Tychonoff (Completely Regular) Property


 * Completely Normal Property

That is, the above properties are all hereditary.

Proof

 * $T_0$ Property is Hereditary


 * $T_1$ Property is Hereditary


 * $T_2$ Property is Hereditary


 * $T_2 \frac 1 2$ Property is Hereditary


 * $T_3$ Property is Hereditary


 * $T_3 \frac 1 2$ Property is Hereditary


 * $T_5$ Property is Hereditary

Proof of Corollary

 * 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.

Note
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.