Topologies on Set form Complete Lattice

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

Let $\mathcal L$ be the set of topologies on $X$.

Then $(\mathcal L, \subseteq)$ is a complete lattice.

Proof
Let $\mathcal K \subseteq \mathcal L$.

Then by Intersection of Topologies is Topology, $\bigcap \mathcal K \in \mathcal L$.

By Intersection is Largest Subset, $\bigcap \mathcal L$ is the infimum of $\mathcal K$.

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

Then $\tau \in \mathcal L$ and $\tau$ is the supremum of $\mathcal K$.

Since each subset of $\mathcal L$ has a supremum and an infimum in $\mathcal L$, $(L, \preceq)$ is a complete lattice.