Topologies on Set form Complete Lattice

Theorem
Let $X$ be a non-empty set.

Let $\LL$ be the set of topologies on $X$.

Then $\struct {\LL, \subseteq}$ is a complete lattice.

Proof
Let $\KK \subseteq \LL$.

Then by Intersection of Topologies is Topology:
 * $\bigcap \KK \in \LL$

By Intersection is Largest Subset, $\bigcap \LL$ is the infimum of $\KK$.

Let $\tau$ be the topology generated by the sub-basis $\bigcup \KK$.

Then $\tau \in \LL$ and $\tau$ is the supremum of $\KK$.

We have that each subset of $\LL$ has a supremum and an infimum in $\LL$.

Thus it follows that $\struct {\LL, \subseteq}$ is a complete lattice.