Separability is not Weakly Hereditary

Theorem

The property of separability is not weakly hereditary.

Proof

It needs to be demonstrated that there exists a separable topological space which has a subspace which is closed but not separable.

Consider an uncountable particular point space $T = \struct {S, \tau_p}$.

From Particular Point Space is Separable, $T$ is separable.

By definition, the particular point $p$ is an open point of $T$.

Thus the subset $S \setminus \set p$ is by definition closed in $T$.

But from Separability in Uncountable Particular Point Space, $S \setminus \set p$ is not separable.

Thus by Proof by Counterexample, separability is not weakly hereditary.

$\blacksquare$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $1$: General Introduction: Countability Properties