Interior Equals Closure of Subset of Discrete Space

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Theorem

Let $T = \left({S, \tau}\right)$ be a discrete topological space.

Let $A \subseteq S$.


Then:

$A = A^\circ = A^-$

where:

$A^\circ$ is the interior of $A$
$A^-$ is the closure of $A$.


Proof

Let $A \subseteq S$.

Then from Set in Discrete Topology is Clopen it follows that $A$ is both open and closed in $T$.

From Closed Set Equals its Closure we have that $A = A^-$.

From Set Interior is Largest Open Set, we have that $A^\circ = A$.

$\blacksquare$


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