Open Set of Uncountable Finite Complement Topology is not F-Sigma

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Let $T = \struct {S, \tau}$ be a finite complement topology on an uncountable set $S$.

Let $U \in \tau$ be an open set of $T$.

Then $U$ is not an $F_\sigma$ set.


Let $U$ be an open set of $T$.

As $S$ is uncountable, then so is $U$.

By the definition of a finite complement topology, all closed sets of $T$ are finite.

From Countable Union of Countable Sets is Countable, $U$ can not be expressed as the union of a countable number of closed sets.

So by definition $U$ is not an $F_\sigma$ set.