Topologies on Set form Complete Lattice

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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.

$\blacksquare$