Characterization of Boundary by Open Sets

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

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

Let $x$ be a point of $T$.

Then $x \in \operatorname{Fr} A$ :
 * for every open set $U$ of $T$:
 * if $x \in U$
 * then $A \cap U \ne \varnothing$ and $\complement_S \left({A}\right) \cap U \ne \varnothing$

where:
 * $\complement_S \left({A}\right) = S \setminus A$ denotes the complement of $A$ in $S$
 * $\operatorname{Fr} A$ denotes the boundary of $A$.

Sufficient Condition
Let $x \in \operatorname{Fr} A$.

Then by Boundary is Intersection of Closure with Closure of Complement:
 * $x \in \left({\complement_S \left({A}\right)}\right)^-$ and $x \in A^-$

where $A^-$ denotes the closure of $A$.

Hence by Condition for Point being in Closure, for every open set $U$ of $T$:
 * $x \in U \implies A \cap U \ne \varnothing$

and:
 * $x \in U \implies \complement_S \left({A}\right) \cap U \ne \varnothing$

Necessary Condition
Let $x$ be such that for every open set $U$ of $T$:
 * if $x \in U$
 * then $A \cap U \ne \varnothing$ and $\complement_S \left({A}\right) \cap U \ne \varnothing$.

Then by Condition for Point being in Closure:
 * $x \in \left({\complement_S \left({A}\right)}\right)^-$ and $x \in A^-$.

Hence by Boundary is Intersection of Closure with Closure of Complement:
 * $x \in \operatorname{Fr} A$