Question

$(X,\tau)$ topolojik uzay olmak üzere $$\emptyset\neq A\subseteq Y\subseteq X\Rightarrow \tau_A=(\tau_Y)_A$$ olduğunu gösteriniz.

Comments

$(\tau_Y)_A=\{U\cap(A\cap Y)\big{|} U\in\tau\}$ 

şeklinde mi hocam?

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Evet. $(\tau_Y)_A=\{U_Y\cap A|U_Y\in \tau_Y\}=\{(U\cap Y)\cap A|U\in \tau\}$

Answer 1

$B\in\tau_A\Rightarrow \begin{array}{cc} \\ \\  \left.\begin{array}{rr} (\exists U\in\tau)(B=U\cap A) \\ \\ \text{Hipotez} \end{array}\right\}  \Rightarrow   (\exists U\in\tau)(B=U\cap (Y\cap A)) \end{array}$

$\\ \Rightarrow (\exists U\in\tau)(B=(U\cap Y)\cap A)$

$\Rightarrow  B\in (\tau_Y)_A \ \  ...(1)$

$B\in (\tau_Y)_A \Rightarrow (\exists U\in\tau)(B=(U\cap Y)\cap A) \Rightarrow \begin{array}{cc} \\ \\  \left.\begin{array}{rr} (\exists U\in\tau)(B=U\cap (Y\cap A)) \\  \\ \text{Hipotez} \end{array}\right\} \Rightarrow \end{array}$

$\Rightarrow (\exists U\in\tau)(B=U\cap A)$

$\Rightarrow B\in\tau_A \ \ ...(2)$

$(1)$ ve $(2) \Rightarrow \tau_A=(\tau_Y)_A$