Interior may not equal Exterior of Exterior/Proof 2

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Theorem

Let $T = \struct {S, \tau}$ be a topological space.

Let $A \subseteq S$ be a subset of the underlying set $S$ of $T$.

Let $A^e$ be the exterior of $A$.

Let $A^\circ$ be the interior of $A$.


Then it is not necessarily the case that:

$A^{ee} = A^\circ$


Proof

Proof by Counterexample:

Let $\struct {S, \preccurlyeq}$ be a totally ordered set.

Let $T = \struct {S, \tau}$ be the Sorgenfrey line on $\struct {S, \preccurlyeq}$.

Let $A \subseteq S$ denote the subset of $S$ defined as:

$A = \openint a {+\infty}$

By Exterior of Exterior in Sorgenfrey Line is not necessarily Interior:

$A^{ee} = \hointr a {+\infty}$

while:

$A^\circ = \openint a {+\infty}$

The result is apparent.

$\blacksquare$


Sources