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.
$\blacksquare$