Separability is not Weakly Hereditary
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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