A subset \(W\) of \(X\) is said to be an “\(F_\sigma\) set” in \(X\) if \(W\) equals a countable union of closed sets of \(X\). Show that \(W\) is an \(F_\sigma\) set in \(X\) if and only if \(X - W\) is a \(G_\delta\) set in \(X\).
[The terminology comes from the French. The “F” stands for “fermé,” which means “closed,” and the “σ” for “somme,” which means “union.”]
Solution
Suppose first that \[ W=\bigcup_{n=1}^{\infty}F_n \] with each \(F_n\) closed. By De Morgan’s law, \[ X-W =X-\bigcup_nF_n =\bigcap_n(X-F_n). \] Each \(X-F_n\) is open, so \(X-W\) is a \(G_\delta\) set.
Conversely, suppose \[ X-W=\bigcap_{n=1}^{\infty}U_n \] with each \(U_n\) open. Again by De Morgan’s law, \[ W =X-\bigcap_nU_n =\bigcup_n(X-U_n). \] Each \(X-U_n\) is closed, so \(W\) is \(F_\sigma\).
Therefore \[ \boxed{W\text{ is }F_\sigma\iff X-W\text{ is }G_\delta.} \]