Complement of F-Sigma Set is G-Delta Set

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Theorem

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

Let $X$ be an $F_\sigma$ set of $T$.


Then its complement $S \setminus X$ is a $G_\delta$ set of $S$.


Proof

Let $X$ be an $F_\sigma$ set of $T$.

Then $X = \ds \bigcup \VV$ where $\VV$ is a countable union of closed sets in $T$.

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

$\ds S \setminus X = S \setminus \bigcup \VV = \bigcap_{V \mathop \in \VV} \paren {S \setminus V}$

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

So $\ds \bigcap_{V \mathop \in \VV} \paren {S \setminus V}$ is a countable intersection of open sets in $T$.

Hence $S \setminus X$ is, by definition, a $G_\delta$ set of $T$.

$\blacksquare$


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