Is the interior of the closure of the interior of the closure of a set equal to the interior of the closure of that set?

Quoting myself from my note here.

As $(\overline{A})^\circ$ is open and a subset of $\overline{(\overline{A})^\circ}$ trivially, by maximality of interior we have indeed $$ (\overline{A})^\circ \subseteq (\overline{(\overline{A})^\circ})^\circ$$

Also $(\overline{A})^\circ \subseteq \overline{A}$, (the interior of a set is a subset of it) this implies (taking the closure on both sides using $\overline{A}$ is closed already) that $\overline{(\overline{A})^\circ} \subseteq \overline{A}$, and then taking the interior on both sides (which preserves the inclusion) gives $$(\overline{(\overline{A})^\circ})^\circ \subseteq \overline{A}^\circ$$ so we have equality $$(\overline{(\overline{A})^\circ})^\circ = \overline{A}^\circ$$