# Separation Properties Preserved in Subspace

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## 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.

### Corollary

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

That is, the above properties are all hereditary.

## Proof

$\blacksquare$

## $T_4$ 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