F-Sigma Sets Closed under Union

Theorem

Let $T = \struct {S, \tau}$ 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 $\sequence {C_n}_{n \mathop \in \N}$ and $\sequence {C'_n}_{n \mathop \in \N}$ of closed sets of $T$ such that:

$F = \ds \bigcup_{n \mathop \in \N} C_n$

$F' = \ds \bigcup_{n \mathop \in \N} C'_n$

By General Self-Distributivity of Set Union, we have:

$F \cup F' = \ds \bigcup_{n \mathop \in \N} \paren {C_n \cup C'_n}$

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

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

$\blacksquare$

Also see

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