F-Sigma Sets form Lattice

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

Let $\FF$ be the collection of all $F_\sigma$ sets of $T$.

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

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

Let $F, F'$ be $F_\sigma$ sets of $T$.

We have $F_\sigma$ Sets Closed under Union, so that $F \cup F' \in \FF$.

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

Similarly, we have $F_\sigma$ Sets Closed under Intersection, and so $F \cap F' \in \FF$.

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

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

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

Also see

 * $G_\delta$ Sets form Lattice