G-Delta Sets Closed under Intersection

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Theorem

Let $T = \struct {S, \tau}$ be a topological space.

Let $G, G'$ be $G_\delta$ sets of $T$.


Then their intersection $G \cap G'$ is also a $G_\delta$ set of $T$.


Proof

By definition of $G_\delta$ set, there exist sequences $\sequence {U_n}_{n \mathop \in \N}$ and $\sequence {U'_n}_{n \mathop \in \N}$ of open sets of $T$ such that:

$G = \displaystyle \bigcap_{n \mathop \in \N} U_n$
$G' = \displaystyle \bigcap_{n \mathop \in \N} U'_n$

By General Distributivity of Intersection, we have:

$G \cap G' = \displaystyle \bigcap_{n \mathop \in \N} \paren {U_n \cap U'_n}$

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

Thus $G \cap G'$ is seen to be a $G_\delta$ set.

$\blacksquare$


Also see