F-Sigma Sets Closed under Intersection

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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.

$\blacksquare$


Also see