$(X,\tau),$ regüler; $(X,\tau), \ T_0$ uzayı; $x,y\in X$ ve $x\neq y$ olsun.
$\left.\begin{array}{rr} (x,y\in X)(x\neq y) \\ \\ (X,\tau), \ T_0\text{ uzayı}\end{array}\right\} \overset{?}{\Rightarrow} \left(y\notin \overline{\{x\}} \vee x\notin \overline{\{y\}}\right)$
I. Durum: $y\notin \overline{\{x\}}$ olsun.
$\left.\begin{array}{rr} y\notin \overline{\{x\}} \\ \\ (X,\tau), \text{ regüler}\end{array}\right\}\Rightarrow \left(\exists U\in\mathcal{U}\left(\overline{\{x\}}\right)\right)(\exists V\in\mathcal{U}(y))(U\cap V=\emptyset)$
$\Rightarrow \left(x\in \overline{\{x\}}\subseteq U\in\tau\right)(V\in\mathcal{U}(y))(U\cap V=\emptyset)$
$\Rightarrow \left(U\in\mathcal{U}(x)\right)(V\in\mathcal{U}(y))(U\cap V=\emptyset).$
II. Durum: $x\notin \overline{\{y\}}$ olsun.
$\left.\begin{array}{rr} x\notin \overline{\{y\}} \\ \\ (X,\tau), \text{ regüler}\end{array}\right\}\Rightarrow (\exists U\in\mathcal{U}\left(\overline{\{y\}}\right)(\exists V\in\mathcal{U}(x))(U\cap V=\emptyset)$
$\Rightarrow \left(y\in \overline{\{y\}}\subseteq U\in\tau\right)(V\in\mathcal{U}(x))(U\cap V=\emptyset)$
$\Rightarrow \left(U\in\mathcal{U}(y)\right)(V\in\mathcal{U}(x))(U\cap V=\emptyset).$
NOT : "?" işaretinin bulunduğu yerdeki geçişin gerekçesine aşağıdaki linkten ulaşabilirsiniz.
https://matkafasi.com/117067/t0-uzaylarinin-karakterizasyonlarina-dair-ii?show=117067#q117067