Let $\{ \mathscr{T}_{\alpha} \}_{\alpha \in \Sigma }$ be a family of topologies on a given set $X$.
Question: How can I find the unique smallest topology on $X$ containing all the collections $\mathscr{T}_{\alpha} $? The unique largest topology contained in all the topologies $\mathscr{T}_{\alpha}$.
Attempt:
I know $\mathscr{L} = \bigcap \mathscr{T}_{\alpha}$ is a topology and is contained in all the topologies $\mathscr{T}_{\alpha} $. How can I show that it is indeed the largest with this property? Also, I don't know what to choose for the smallest topology.
thanks for any help.
Best Answer
Concerning your attempt:
Suppose that $\tau$ is a topology that is contained in $\mathcal{T_{\alpha}}$ for each $\alpha\in\Sigma$. So if $U\in\tau$ then $U\in\mathcal{T_{\alpha}}$ for each $\alpha\in\Sigma$ or equivalently $U\in\mathcal{L}:=\bigcap_{\alpha\in\Sigma}\mathcal{T_{\alpha}}$. This proves that $\tau\subseteq\mathcal{L}$ so confirms that $\mathcal{L}$ is the 'largest' with the mentioned property.
Concerning the smallest topology containing all topologies $\mathcal T_{\alpha}$.
Defining $\mathcal{W}=\bigcup_{\alpha\in\Sigma}\mathcal{T_{\alpha}}$. you are looking for the smallest topology $\mathcal{O}$ that contains $\mathcal{W}$. Then $\mathcal{O}$ must contain the collection: $$\mathcal{W}':=\left\{ W\mid W\text{ is a finite intersection of elements in }\mathcal{W}\right\} $$ The collection $\mathcal W'$ can be shown to be a basis for a topology. Based on this we conclude that $\mathcal{O}$ must contain the collection: $$\mathcal{W}'':=\left\{ W\mid W\text{ is a union of elements in }\mathcal{W}'\right\} $$
and that $\mathcal{W}''$ is a topology. So we come to $\mathcal{O}=\mathcal{W}''$.
In fact this way any collection $\mathcal{V}\subseteq\wp\left(X\right)$ gives rise to a topology $\mathcal{O}_{\mathcal{V}}$ that is constructed as described above (and identifying the 'empty intersection' with $X$). This $\mathcal{O}_{\mathcal{V}}$ is the topology generated by $\mathcal{V}$.