G-Delta Sets form Lattice

Theorem
Let $T = \struct {S, \tau}$ be a topological space.

Let $\GG$ be the collection of all $G_\delta$ sets of $T$.

Then $\struct {\GG, \subseteq}$ is a lattice, where $\subseteq$ denotes the subset relation.

Proof
From Subset Relation is Ordering, $\subseteq$ is an ordering on $\GG$.

Let $G, G'$ be $G_\delta$ sets of $T$.

We have $G_\delta$ Sets Closed under Union, so that $G \cup G' \in \GG$.

From Union is Smallest Superset and Subset of Union, it follows that $G \cup G'$ is the supremum of $G$ and $G'$.

Similarly, we have $G_\delta$ Sets Closed under Intersection, and so $G \cap G' \in \GG$.

From Intersection is Largest Subset and Intersection is Subset, it follows that $G \cap G'$ is the infimum of $G$ and $G'$.

Thus any two elements of $\GG$ are seen to have both a supremum and an infimum in $\GG$.

Hence $\struct {\GG, \subseteq}$ is a lattice.

Also see

 * $F_\sigma$ Sets form Lattice