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 $\KK$.
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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.
$\blacksquare$