Characterization of Closure by Open Sets

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

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

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

Let $A^-$ denote the closure of $A$.

Then $x \in A^-$ :
 * for every open set $U$ of $T$:
 * $x \in U \implies A \cap U \ne \varnothing$

Sufficient Condition
Let $x \in A^-$.

there exists an open set $U$ of $T$ such that:
 * $x \in U$ and $A \cap U = \varnothing$

We have that $U$ is open in $T$.

So by definition of closed set, $\complement_S \left({U}\right)$ is closed in $T$.

Then:

But we have:
 * $x \in A^-$

and also:
 * $x \in U$

and thus by definition of set intersection:
 * $x \in A^- \cap U$

This contradicts $A^- \cap U = \varnothing$

Hence by Proof by Contradiction the assumption that there exists an open set $U$ of $T$ such that $x \in U$ and $A \cap U = \varnothing$ was false.

So for every open set $U$ of $T$:
 * $x \in U \implies A \cap U \ne \varnothing$

Necessary Condition
Let $x$ be such that for every open set $U$ of $T$:
 * $x \in U \implies A \cap U \ne \varnothing$

$x \notin A^-$.

Then:
 * $x \in \complement_S \left({A^-}\right)$

Then by assumption:
 * $A \cap \complement_S \left({A^-}\right) \ne \varnothing$

By definition of complement:
 * $A \cap \complement_S \left({A}\right) = \varnothing$

So by Empty Intersection iff Subset of Complement:
 * $A \not \subseteq A^-$

From Set is Subset of its Topological Closure:
 * $A \subseteq A^-$

But from Set Complement inverts Subsets:
 * $\complement_S \left({A^-}\right) \subseteq \complement_S \left({A}\right)$

from which by Empty Intersection iff Subset of Complement:
 * $A \cap \complement_S \left({A^-}\right) = \varnothing$

Hence by Proof by Contradiction the assumption that $x \notin A^-$ was false.

So $x \in A^-$.