Union of Interiors and Boundary Equals Whole Space

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Theorem

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

Let $A$ be a subset of $T$.


Then:

$S = \operatorname{Int} A \cup \operatorname{Fr} A \cup \operatorname{Int} A'$

where:

$A' = S \setminus A$ denotes the complement of $A$ relative to $S$
$\operatorname{Int} A$ denotes the interior of $A$
$\operatorname{Fr} A$ denotes the boundary of $A$.


Proof

\(\displaystyle \operatorname{Int} A \cup \operatorname{Fr} A \cup \operatorname{Int} A'\) \(=\) \(\displaystyle \operatorname{Int} A \cup \operatorname{Int} A' \cup \operatorname{Fr} A\) Union is Associative, Union is Commutative
\(\displaystyle \) \(=\) \(\displaystyle \operatorname{Int} A \cup \operatorname{Int} A' \cup \left({\cl A \cap \cl A'}\right)\) Boundary is Intersection of Closure with Closure of Complement
\(\displaystyle \) \(=\) \(\displaystyle \left({\operatorname{Int} A \cup \operatorname{Int} A' \cup \cl A}\right) \cap \left({\operatorname{Int} A \cup \operatorname{Int} A' \cup \cl A'}\right)\) Intersection Distributes over Union
\(\displaystyle \) \(=\) \(\displaystyle \left({\operatorname{Int} A \cup \left({\cl A}\right)' \cup \cl A}\right) \cap \left({\operatorname{Int} A \cup \operatorname{Int} A' \cup \cl A'}\right)\) Complement of Interior equals Closure of Complement
\(\displaystyle \) \(=\) \(\displaystyle \left({\operatorname{Int} A \cup \left({\left({\cl A}\right)' \cup \cl A}\right)}\right) \cap \left({\operatorname{Int} A \cup \operatorname{Int} A' \cup \cl A'}\right)\) Union is Associative
\(\displaystyle \) \(=\) \(\displaystyle \left({\operatorname{Int} A \cup S}\right) \cap \left({\operatorname{Int} A \cup \operatorname{Int} A' \cup \cl A'}\right)\) Union with Relative Complement
\(\displaystyle \) \(=\) \(\displaystyle \left({\operatorname{Int} A \cup S}\right) \cap \left({\operatorname{Int} A \cup \operatorname{Int} A' \cup \left({\operatorname{Int} A}\right)'}\right)\) Complement of Interior equals Closure of Complement
\(\displaystyle \) \(=\) \(\displaystyle \left({\operatorname{Int} A \cup S}\right) \cap \left({\operatorname{Int} A \cup \left({\operatorname{Int} A}\right)' \cup \operatorname{Int} A'}\right)\) Union is Associative, Union is Commutative
\(\displaystyle \) \(=\) \(\displaystyle \left({\operatorname{Int} A \cup S}\right) \cap \left({S \cup \operatorname{Int} A'}\right)\) Union with Relative Complement
\(\displaystyle \) \(=\) \(\displaystyle S \cap \left({S \cup \operatorname{Int} A'}\right)\) Union with Superset is Superset
\(\displaystyle \) \(=\) \(\displaystyle S \cap S\) Union with Superset is Superset
\(\displaystyle \) \(=\) \(\displaystyle S\) Intersection is Idempotent

$\blacksquare$


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