Topoloji sorusu

Question

$(X ,\tau)$ topolojik uzay olmak üzere, 

$\tau=C(X, \tau)\Rightarrow (X, \tau)$, regüler uzay.

$(X, \tau), \text{regüler uzay}:\Leftrightarrow(\forall F\in C(X,\tau)) [x\notin F\Rightarrow (\exists U\in\mathcal {U}(F)) (\exists V\in\mathcal{U}(x)) (U\cap V=\emptyset)]$

Answer 1

$\tau=\mathcal{C}(X,\tau);F\in\mathcal{C}(X,\tau)$ ve $x\notin F \text{olsun.}$

$(F\in\mathcal{C}(X,\tau))(x\notin F)\Rightarrow\left.\begin{array}{rr} (F\in\mathcal{C}(X,\tau))(x\in\setminus F)  \\ F\in\mathcal{C}(X,\tau)\Rightarrow\setminus F\in\tau\end{array}\right\}\Rightarrow$

$\Rightarrow\left.\begin{array}{rr} (F\in\mathcal{C}(X,\tau))(x\in\setminus F\in\tau)  \\ \tau=\mathcal{C}(X,\tau)\end{array}\right\}\Rightarrow (U:=F\in\tau) (V:=\setminus F\in\tau)$

$\Rightarrow (U\in\mathcal{U}(F)) (V\in\mathcal{U}(x)) (U\cap V=\emptyset)$