Kompakt ve Hausdorff olan her topolojik uzayın bir regüler uzay olduğunu gösteriniz.

Question

$(X,\tau)$ topolojik uzay olmak üzere

$$((X,\tau), \text{ kompakt uzay})((X,\tau), \text{ Hausdorff})\Rightarrow (X,\tau), \text{ regüler}$$ olduğunu gösteriniz.

Answer 1

$(X,\tau ),$ kompakt uzay; $\left( X,\tau \right) ,$ Hausdorff; $A\in \mathcal{C}\left( X,\tau \right)$ ve $x\notin A$ olsun.$\left.\begin{array}{r} x\notin A \\ \\ y\in A\end{array}\right\} \Rightarrow \begin{array}{c} \mbox{} \\ \mbox{} \\ \left.\begin{array}{c} x\neq y \\ \mbox{ } \\ {\left( X,\tau \right) ,\text{ }T_{2}} \end{array} \right\} \Rightarrow \!\! \begin{array}{c} \mbox{} \\ \mbox{} \\ \left. \begin{array}{r} \left( \exists U_{x}\in \mathcal{U}\left( x\right) \right) \left( \exists V_{y}\in \mathcal{U}\left( y\right) \right) \left( U_{x}\cap V_{y}=\emptyset \right)  \\ \\ \mathcal{A}:=\left\{ V_{y}|y\in A\right\} \end{array}\right\} \Rightarrow \end{array}\end{array}$

$\mbox{}$

$\left.\begin{array}{r} \Rightarrow \left( \mathcal{A}\subseteq \tau \right) \left( A\subseteq \cup \mathcal{A}\right)\\ \\ \left( (X,\tau ),\text{ kompakt uzay}\right) \left( A\in \mathcal{C}\left( X,\tau\right) \right) \Rightarrow A,\text{ }\tau \text{-kompakt}\end{array}\right\} \Rightarrow$

$\mbox{}$

$\left.\begin{array}{r}\Rightarrow \left( \exists \mathcal{A}^{\ast }\subseteq \mathcal{A}\right) \left( \left\vert \mathcal{A}^{\ast}\right\vert <\aleph _{0}\right) \left( A\subseteq \cup \mathcal{A}^{\ast}\right) \\ \\ \left( U:=\cup \mathcal{A}^{\ast }\right) (V:=\setminus A)\end{array}\right\} \Rightarrow \left( U\in \mathcal{U}(A)\right)\left( V\in \mathcal{U}\left(x\right) \right) \left( U\cap V=\emptyset \right).$