Topolojik Uzaylarda Kapanışa Dair

Question

$(X,\tau)$ topolojik uzay ve $A,B\subseteq X$ olmak üzere

$$A\in\tau \Rightarrow A\cap \overline{B}\subseteq \overline{A\cap B}$$ olduğunu gösteriniz.

Answer 1

$A\in \tau$ ve $x\in A\cap\overline{B}$ olsun.

$$\left.\begin{array}{rr} x\in A\cap\overline{B}\Rightarrow(x\in A)\left(x\in \overline{B}\right)\Rightarrow(x\in A)(\forall U\in\mathcal{U}(x))(U\cap B\neq \emptyset)\\ A\in\tau \end{array}\right\}\Rightarrow$$

$$\Rightarrow$$

$$(\forall U\in\mathcal{U}(x))(U\cap A\in\mathcal{U}(x))((U\cap A)\cap B\neq \emptyset)$$

$$\Rightarrow$$

$$(\forall U\in\mathcal{U}(x))(U\cap (A\cap B)\neq \emptyset)$$

$$\Rightarrow$$

$$x\in \overline{A\cap B}.$$