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## Homework Statement

I am working on the proof that taking closure and interior of a set in a metric space can produce at most 7 sets. The piece I need is that [itex] \bar{\mathring{A}} = \bar{\mathring{\bar{\mathring{A}}}} [/itex].

## Homework Equations

Interior of A is the union of all open sets contained in A, aka the largest open set contained in A.

Closure of A is the intersection of all closed sets containing A, aka the smallest closed set containing A.

## The Attempt at a Solution

[itex] \bar{\mathring{A}}[/itex] is a closed set. [itex]\mathring{\bar{\mathring{A}}}\subseteq \bar{\mathring{A}}[/itex]. Since [itex]\bar{\mathring{\bar{\mathring{A}}}} [/itex] is the smallest closed set containing [itex] \mathring{\bar{\mathring{A}}} [/itex] we have that [itex]\bar{\mathring{\bar{\mathring{A}}}}\subseteq \bar{\mathring{A}}[/itex].

I'm not sure how to get the inclusion [itex] \bar{\mathring{A}}\subseteq \bar{\mathring{\bar{\mathring{A}}}} [/itex]

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