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Let $\{T_\alpha\}$ be a family of topologies on $X$. Show that there is a unique smallest topology on $X$ containing all the collections $T_\alpha$, and a unique largest topology contained in all $T_\alpha$.

We can check that $\bigcap T_\alpha$ is a topology on $X$, so it is the unique largest topology contained in all $T_\alpha$.

Now, a topology containing all $T_\alpha$ must contain $\bigcup T_\alpha$. It must also contain arbitrary unions and finite intersections of sets in $\bigcup T_\alpha$. Since the union of the sets in $T_\alpha$ is $X$, this is the topology generated by the subbasis $\bigcup T_\alpha$. How can we prove that it is the unique smallest one containing all the collections $T_\alpha$?

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Let $\sigma _1$ be the topology generated by $\cup T_{\alpha}$ as a subbasis. Verify that any topology $\sigma_2$ on $X$ that contains $\cup T_{\alpha}$ must satisfy $\sigma_1\subseteq \sigma_2$.

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