Hausdorff Uzaylarının Karakterizasyonuna Dair-II

Question

$(X,\tau)$ topolojik uzay olmak üzere

$$(X,\tau), \text{ Hausdorff}\Leftrightarrow (\forall x,y\in X)[x\neq y\Rightarrow (\exists U\in\mathcal{U}(x))(y\notin\overline{U})].$$

Answer 1

$(\Rightarrow):$ $(X,\tau), \ T_2$ uzayı; $x,y\in X$ ve $x\neq y$ olsun.

$\left.\begin{array}{rr} (x,y\in X)(x\neq y) \\ \\ (X,\tau), \ T_2\text{ uzayı} \end{array}\right\}\Rightarrow (\exists U\in\mathcal{U}(x))(\exists V\in\mathcal{U}(y))(U\cap V=\emptyset)\Rightarrow (\exists U\in\mathcal{U}(x))(y\notin \overline{U}).$

$(\Leftarrow):$ $x,y\in X$ ve $x\neq y$ olsun.

$\left.\begin{array}{rr} (x,y\in X)(x\neq y) \\ \\ \text{Hipotez} \end{array}\right\}\Rightarrow (\exists U\in\mathcal{U}(x))(y\notin \overline{U})\Rightarrow (\exists U\in\mathcal{U}(x))(\exists V\in\mathcal{U}(y))(U\cap V=\emptyset).$