F-Sigma Sets Closed under Union

Theorem
Let $T = \left({S, \vartheta}\right)$ be a topological space.

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

Then their union $F \cup F'$ is also an $F_\sigma$ set of $T$.

Proof
By definition of $F_\sigma$ set, there exist sequences $\left({C_n}\right)_{n \in \N}$ and $\left({C'_n}\right)_{n \in \N}$ of closed sets of $T$ such that:


 * $F = \displaystyle \bigcup_{n \mathop \in \N} C_n$
 * $F' = \displaystyle \bigcup_{n \mathop \in \N} C'_n$

By General Distributivity of Set Union, we have:


 * $F \cup F' = \displaystyle \bigcup_{n \mathop \in \N} \left({C_n \cup C'_n}\right)$

By Finite Union of Closed Sets is Closed, $C_n \cup C'_n$ is closed, for all $n \in \N$.

Thus $F \cup F'$ is seen to be an $F_\sigma$ set.

Also see

 * $F_\sigma$ Sets Closed under Intersection
 * $G_\delta$ Sets Closed under Union