Complement of G-Delta Set is F-Sigma Set

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Theorem

Let $T = \struct {S, \tau}$ 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 $X = \ds \bigcap \UU$ where $\UU$ is a countable intersection of open sets in $T$.

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

$\ds S \setminus X = S \setminus \bigcap \UU = \bigcup_{U \mathop \in \UU} \paren {S \setminus U}$

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

So $\ds \bigcup_{U \mathop \in \UU} \paren {S \setminus U}$ 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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