# F-Sigma Sets Closed under Intersection

## Theorem

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

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

Then their intersection $F \cap F'$ is also a $F_\sigma$ 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:

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

Now compute:

 $\displaystyle F \cap F'$ $=$ $\displaystyle \bigcup_{n \mathop \in \N} \left({C_n \cap F'}\right)$ Intersection Distributes over Union: General Result $\displaystyle$ $=$ $\displaystyle \bigcup_{n \mathop \in \N} \bigcup_{m \mathop \in \N} \left({C_n \cap C'_m}\right)$ Intersection Distributes over Union: General Result

By Intersection of Closed Sets is Closed, $C_n \cap C'_m$ is closed, for all $n, m \in \N$.

By Cartesian Product of Countable Sets is Countable, $\N \times \N$ is countable.

Thus $F \cap F'$ is seen to be a countable union of closed sets, hence a $F_\sigma$ set.

$\blacksquare$