Question

$(X,d)$ metrik uzay, $A\subseteq X$ ve $x\in X$ olmak üzere $$x\in\overline{A}\Leftrightarrow (\forall\epsilon>0)(\exists y\in A)(d(x,y)<\epsilon)$$ olduğunu gösteriniz.

Answer 1

Kanıt: $(\Rightarrow):$ $x\in\overline{A}$  ve  $\epsilon>0$ olsun.

$\left.\begin{array}{rr} x\in\overline{A} \\ \\ \epsilon>0\end{array}\right\}\Rightarrow B(x,\epsilon)\cap A\neq\emptyset\Rightarrow(\exists y\in X)(y\in B(x,\epsilon)\cap A)\Rightarrow(y\in X)(y\in A)(y\in B(x,\epsilon))$

$\left.\begin{array}{rr} \Rightarrow(y\in X\cap A)(y\in B(x,\epsilon)) \\ \\ A\subseteq X\Rightarrow A\cap X=A\end{array}\right\}\Rightarrow (y\in A)(d(x,y)<\epsilon).$

$(\Leftarrow):$ $\epsilon>0$ olsun.

$\left.\begin{array}{rr} \epsilon>0 \\ \\ \text{Hipotez} \end{array}\right\}\Rightarrow (\exists y\in A)(d(x,y)<\epsilon)\Rightarrow (y\in A)(y\in B(x,\epsilon))\Rightarrow y\in B(x,\epsilon)\cap A\Rightarrow B(x,\epsilon)\cap A\neq\emptyset$

elde edilir. O halde $x\in\overline{A}$ olur.