$\{T_a\}$ is a collection of topologies in $X$, show that there exists a least and bigger topology in $X$ There's a question in my book that says the following:
$\{T_a\}$ is a collection of topologies in $X$, show that $\cap T_a$ is a topology in $X$. What about $\cup T_a$? Also, show that there exists an unique "least" topology in $X$ containing all the collections $T_a$ and a bigger topology contained in all $T_a$.
I did the part of the union and intersection of topologies, but I don't even know how to begin in the last part of the question. What is a 'least' topology that contains $T_a$? For me, this would be the biggest one, and for the one that is contained in all $T_a$, this would be the least one. This doesn't even makes sense. Could somebody tell me what the exercise is asking and how to prove it?
 A: Let $\mathscr{T}=\{\tau_\alpha:\alpha\in A\}$ be a collection of topologies on a set $X$. Let $\mathscr{B}=\bigcup\mathscr{T}=\bigcup_{\alpha\in A}\tau_\alpha$, the union of all of these topologies.


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*Show that $\mathscr{B}$ is a base for some topology.


Call that topology $\tau$.


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*Show that $\tau_\alpha\subseteq\tau$ for each $\alpha\in A$.  

*Show further that if $\tau'$ is any topology on $X$ such that $\tau_\alpha\subseteq\tau'$ for all $\alpha\in A$, then $\tau\subseteq\tau'$. This shows that $\tau$ is the smallest topology containing all of the topologies in $\mathscr{T}$: it’s contained in every topology that contains every member of $\mathscr{T}$.  

*Explain why $\tau$ is the only topology on $X$ with these properties.


For the rest of the problem let $\tau=\bigcap\mathscr{T}=\bigcap_{\alpha\in A}\tau_\alpha$, the intersection of all of these topologies. You’ve already shown that $\tau$ is a topology on $X$, and it’s clear that $\tau\subseteq\tau_\alpha$ for each $\alpha\in A$. To finish the problem, you need to show that $\tau$ is the largest topology on $X$ that is contained in each of the topologies $\tau_\alpha$. That is, you need to show this:


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*If $\tau'$ is a topology on $X$, and $\tau'\subseteq\tau_\alpha$ for each $\alpha\in A$, then $\tau'\subseteq\tau$. In words, every topology on $X$ that is a subset of each of the topologies in $\mathscr{T}$ is also a subset of $\tau$, so $\tau$ is the largest topology that is contained in each $\tau_\alpha$ with $\alpha\in A$.

