$(\Rightarrow):$ $(X,\tau), \ T_0$ uzayı; $x,y\in X$ ve $x\neq y$ olsun.
$\left.\begin{array}{rr} (x,y\in X)(x\neq y) \\ \\ (X,\tau), \ T_0 \text{ uzayı}\end{array}\right\}\Rightarrow (\exists U\in\mathcal{U}(x))(y\notin U)\vee (\exists V\in\mathcal{U}(y))(x\notin V)$
$\Rightarrow (\exists U\in\mathcal{U}(x))(U\cap \{y\}=\emptyset) \vee (\exists V\in\mathcal{U}(y))(V\cap \{x\}=\emptyset)$
$\Rightarrow x\notin \overline{\{y\}} \vee y\notin\overline{\{y\}}$
I. Durum: $x\notin \overline{\{y\}}$ olsun.
$\left.\begin{array}{rr} x\in X\Rightarrow \{x\}\subseteq \overline{\{x\}}\Rightarrow x\in \overline{\{x\}} \\ \\ x\notin \overline{\{y\}} \end{array}\right\}\Rightarrow \overline{\{x\}}\neq\overline{\{y\}}.$
II. Durum: $y\notin \overline{\{x\}}$ olsun.
$\left.\begin{array}{rr} y\in X\Rightarrow \{y\}\subseteq \overline{\{y\}}\Rightarrow y\in \overline{\{y\}} \\ \\ y\notin \overline{\{x\}} \end{array}\right\}\Rightarrow \overline{\{x\}}\neq\overline{\{y\}}.$
$(\Leftarrow):$ $x,y\in X$ ve $x\neq y$ olsun. $x$ noktasının $y$ noktasını içermeyen veya $y$ noktasının $x$ noktasını içermeyen en az bir açık komşuluğunun var olduğunu gösterirsek ispat biter.
$\left.\begin{array}{rr} (x,y\in X)(x\neq y) \\ \\ \text{Hipotez}\end{array}\right\}\Rightarrow \overline{\{x\}}\neq\overline{\{y\}}\Rightarrow \left(\overline{\{x\}}\setminus \overline{\{y\}}\neq \emptyset \vee \overline{\{y\}}\setminus \overline{\{x\}}\neq \emptyset\right)$
$\Rightarrow \left(\exists u\in X\right)\left(u\in \overline{\{x\}}\setminus \overline{\{y\}}\right)\vee \left(\exists v\in X\right)\left(v\in \overline{\{y\}}\setminus \overline{\{x\}}\right)$
$\Rightarrow \left(u\in \overline{\{x\}}\right)\left(u\notin \overline{\{y\}}\right) \vee \left(v\in \overline{\{y\}}\right)\left(v\notin \overline{\{x\}}\right)$
$\Rightarrow \left(\exists U\in\mathcal{U}(u)\right)(U\cap \{y\}=\emptyset)(U\cap \{x\}\neq\emptyset) \vee \left(\exists V\in\mathcal{U}(v)\right)(V\cap \{x\}=\emptyset)(V\cap \{y\}\neq\emptyset) $
$\Rightarrow \left(U\in\mathcal{U}(x)\right)(y\notin U) \vee \left(V\in\mathcal{U}(y)\right)(x\notin V).$