Question

$(X,\tau)$ topolojik uzay ve $\emptyset\neq Y\subseteq X$ olmak üzere $$\tau_Y:=\{Y\cap T|T\in\tau\}$$ ailesinin $Y$ kümesi üzerinde bir topoloji olduğunu gösteriniz.

Answer 1

$T_1) \mbox{ } \emptyset,Y\overset{?}{\in}\tau_Y$

$$\left.\begin{array}{rr} \emptyset\in\tau\Rightarrow Y\cap \emptyset\in\tau_Y \\ Y\cap \emptyset=\emptyset \end{array}\right\}\Rightarrow \emptyset\in\tau_Y \,\,\,\,\,\,\,\,\,\,\,\,\ \left.\begin{array}{rr} X\in\tau\Rightarrow Y\cap X\in\tau_Y \\ Y\subseteq X\Rightarrow Y\cap X=Y \end{array}\right\}\Rightarrow Y\in\tau_Y$$

$T_2) \mbox{ } A,B\in\tau_Y$  olsun.

$$\left.\begin{array}{rr} A\in\tau_Y\Rightarrow (\exists T_1\in\tau)(A=Y\cap T_1) \\ B\in\tau_Y\Rightarrow (\exists T_2\in\tau)(B=Y\cap T_2) \end{array}\right\}\Rightarrow (T_1\cap T_2\in\tau)(A\cap B=(Y\cap T_1)\cap(Y\cap T_2))$$

$$\Rightarrow$$

$$(T_1\cap T_2\in\tau)(A\cap B=Y\cap(T_1\cap T_2))$$

$$\Rightarrow$$

$$A\cap B\in \tau_Y.$$

$T_3) \mbox{ } \mathcal{A}_Y\subseteq \tau_Y$ olsun.

$$\mathcal{A}_Y\subseteq \tau_Y$$

$$\Rightarrow$$

$$\mathcal{A}:=\{T|A\in\mathcal{A}_Y\Rightarrow (\exists T\in\tau) (A=Y\cap T)\}\subseteq \tau$$

$$\Rightarrow$$

$$(\cup \mathcal{A} \in\tau)\left(\cup \mathcal{A}_Y=\bigcup_{A\in\mathcal{A}_Y}A=\bigcup_{T\in\mathcal{A}}(Y\cap T)=Y\cap\left(\bigcup_{T\in\mathcal{A}}T\right)=Y\cap(\cup\mathcal{A})\right)$$

$$\Rightarrow$$

$$\cup \mathcal{A}_Y\in\tau_Y.$$