# F-Sigma Sets Closed under Intersection

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## Theorem

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

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

Now compute:

\(\ds F \cap F'\) | \(=\) | \(\ds \bigcup_{n \mathop \in \N} \paren {C_n \cap F'}\) | Intersection Distributes over Union: General Result | |||||||||||

\(\ds \) | \(=\) | \(\ds \bigcup_{n \mathop \in \N} \bigcup_{m \mathop \in \N} \paren {C_n \cap C'_m}\) | Intersection Distributes over Union: General Result |

By Intersection of Closed Sets is Closed in Topological Space, $C_n \cap C'_m$ 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.

$\blacksquare$