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 $T = \left({S, \tau_p}\right)$.

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

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

Thus the subset $S \setminus \left\{{p}\right\}$ is by definition closed in $T$.

But from Separability in Uncountable Particular Point Space, $S \setminus \left\{{p}\right\}$ is not separable.

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