Separation Properties Preserved in Subspace

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:

$T_0$ (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

Completely Hausdorff Property is Hereditary

$T_3$ Property is Hereditary

$T_3 \frac 1 2$ Property is Hereditary

$T_5$ Property is 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