Characterization of T0 Space by Distinct Closures of Singletons

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

Then
 * $T$ is a $T_0$ space
 * $\forall x, y \in S: x \ne y \implies \left\{{x}\right\}^- \ne \left\{{y}\right\}^-$

where $\left\{{y}\right\}^-$ denotes the closure of $\left\{{y}\right\}$.

Sufficient Condition
Let $T$ be a $T_0$ space.

Let $x, y \in S$ such that
 * $x \ne y$

By definition of $T_0$ space:
 * $\left({\exists U \in \tau: x \in U \land y \notin U}\right) \lor \left({\exists U \in \tau: x \notin U \land y \in U}\right)$

WLOG: Suppose
 * $\exists U \in \tau: x \in U \land y \notin U$

By definition of singleton:
 * $x \in \left\{{x}\right\}$

By Set is Subset of its Topological Closure:
 * $\left\{{x}\right\} \subseteq \left\{{x}\right\}^-$

Then by definition of subset:
 * $x \in \left\{{x}\right\}^-$

Hence by definition of intersection:
 * $\left\{{x}\right\}^- \cap U \ne \varnothing$

By definition of singleton:
 * $\forall z: z \in \left\{{y}\right\} \implies z = y$

Then by definition of intersection:
 * $\left\{{y}\right\} \cap U = \varnothing$

Hence by Disjoint Open Sets remain Disjoint with one Closure
 * $\left\{{y}\right\}^- \cap U = \varnothing$

Thus:
 * $\left\{{x}\right\}^- \ne \left\{{y}\right\}^-$

Necessary Condition
Assume that:
 * $(1): \quad \forall x, y \in S: x \ne y \implies \left\{{x}\right\}^- \ne \left\{{y}\right\}^-$

By Characterization of $T_0$ Space by Closed Sets it suffices to prove that
 * $\left({\exists F \subseteq S: F}\right.$ is closed $\left. {\land x \in F \land y \notin F}\right)$

or
 * $\left({\exists F \subseteq S: F}\right.$ is closed $\left. {\land x \notin F \land y \in F}\right)$

Then assume:
 * $(2): \quad \forall F \subseteq S: F$ is closed $\land x \in F \implies y \in F$

Define $F := \left\{{y}\right\}^-$

By Topological Closure is Closed:
 * $F$ is closed.

We will prove now that
 * $x \notin F$


 * $x \in F$

Then:
 * $\left\{{x}\right\} \subseteq F$

By Topological Closure of Subset is Subset of Topological Closure:
 * $\left\{{x}\right\}^- \subseteq F^-$

By Set is Closed iff Equals Topological Closure:
 * $(3): \quad \left\{{x}\right\}^- \subseteq F$

By definition of singleton:
 * $x \in \left\{{x}\right\}$

By Set is Subset of its Topological Closure:
 * $\left\{{x}\right\} \subseteq \left\{{x}\right\}^-$

Then by definition of subset:
 * $x \in \left\{{x}\right\}^-$

Because $\left\{{x}\right\}^-$ is closed therefore by $(2)$:
 * $y \in \left\{{x}\right\}^-$

Then:
 * $\left\{{y}\right\} \subseteq \left\{{x}\right\}^-$

By Topological Closure of Subset is Subset of Topological Closure:
 * $F \subseteq \left({\left\{{x}\right\}^-}\right)^-$

Then by Closure of Topological Closure equals Closure:
 * $F \subseteq \left\{{x}\right\}^-$

Hence by $(3)$ and definition of set equality:
 * $F = \left\{{x}\right\}^-$

This contradicts the assumption $(1)$.

Thus $x \notin F$.

By definition of singleton:
 * $y \in \left\{{y}\right\}$

By Set is Subset of its Topological Closure:
 * $\left\{{y}\right\} \subseteq \left\{{y}\right\}^- = F$

Thus by definition of subset:
 * $y \in F$

Hence
 * $\exists F \subseteq S: F$ is closed $\land x \notin F \land y \in F$