On the largest and smallest topology on a given set.
In drhab's answer, he defines $\mathcal{W}=\bigcup_{\alpha\in\Sigma}\mathcal{T_{\alpha}}.$ $$\mathcal{W}':=\left\{ W\mid W\text{ is a finite intersection of elements in }\mathcal{W}\right\} .$$ Then defines $$\mathcal{W}'':=\left\{ W\mid W\text{ is a union of elements in }\mathcal{W}'\right\} .$$ Then, proves $$\mathcal{W}'' $$ is a topology on $X$. What happens if we defines otherway? Then defines $$\mathcal{T}':=\left\{ W\mid W\text{ is a union of elements in }\mathcal{W}\right\} .$$ Then, $$\mathcal{T}'':=\left\{ W\mid W\text{ is a finite intersection of elements in }\mathcal{T'}\right\} .$$ Will $\mathcal{T}''$ be a topology on $X$? If so, Will $\mathcal{T}''=\mathcal{W}''$?
My attempt to prove $\mathcal{T}''$ be a topology on $X$
(1) $X\in \mathcal{T}'' $ and $\phi \in \mathcal{T}''$ are trivial since $,\phi,X\in \mathcal{T_\alpha}.$
(2)$X_1,X_2\in \mathcal{T}''$. Then, $X_1=\bigcap_{i=1}^k B_i$, $B_i\in \mathcal{T}'$ and $X_2=\bigcap_{j=1}^l C_j,$$C_j\in \mathcal{T}'$. Hence, by definition of $ \mathcal{T}''$, $X_1 \cap X_2 \in \mathcal{T}''$.
(3) Let $X_{\alpha}\in \mathcal{T}''$. $X_{\alpha}=\bigcap_{j=1}^l D_j^{\alpha},$$D_j^{\alpha}\in \mathcal{T}'. $ Then, $\bigcup_{\alpha}X_{\alpha}=\bigcup_{\alpha\in \Lambda}\bigcap_{j=1}^l D_j^{\alpha}$. where, $D_j^{\alpha}=\bigcup_{\beta \in \Gamma} E_{\alpha,j}^{\beta}$. How do I proceed further?