Complement of G-Delta Set is F-Sigma Set

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Theorem

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

Let $X$ be an $G_\delta$ set of $T$.


Then its complement $S \setminus X$ is an $F_\sigma$ set of $T$.


Proof

Let $X$ be a $G_\delta$ set of $T$.

Let $X = \displaystyle \bigcap \mathcal U$ where $\mathcal U$ is a countable intersection of open sets in $T$.

Then from De Morgan's Laws: Difference with Intersection we have:

$\displaystyle S \setminus X = S \setminus \bigcap \mathcal U = \bigcup_{U \mathop \in \mathcal U} \left({S \setminus U}\right)$

By definition of closed set, each of the $S \setminus U$ are closed sets.

So $\displaystyle \bigcup_{U \mathop \in \mathcal U} \left({S \setminus U}\right)$ is a countable union of closed sets in $T$.

Hence $S \setminus X$ is, by definition, an $F_\sigma$ set of $T$.

$\blacksquare$


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