# Union of Interiors and Boundary Equals Whole Space

## 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

$\blacksquare$