Exercise 3.5.7

For a set $A\subseteq R$ define $-A=\{-x:x\in A\}$. Note that if $A$ is closed (open), then so is $-A$, and if $A$ is a $F_{\sigma }$ set ( $G_{\delta }$), so is $-A$.

Define $I^{\text{+}}=I\cap [0,\infty )$, $I^{\text{−}}=I\cap (-\infty ,0]$ , and $Q^{\text{+}}$ and $Q^{\text{−}}$ be defined similarly for $Q$. Consider the set $A=I^{\text{+}}\cup Q^{\text{−}}$. If $A$ is a $F_{\sigma }$ set, then by Exercise 3.5.2 so is $A\cap [0,\infty )=I^{\text{+}}$, but so is $I^{\text{+}}\cup -I^{\text{+}}=I^{\text{+}}\cup I^{\text{−}}=I$, which is a contradiction. Similarly, $A$ being a $G_{\delta }$ set implies $A\cap (-\infty ,0]=Q^{\text{−}}$ and $Q^{\text{−}}\cup -Q^{\text{−}}=Q$ are both $G_{\delta }$ sets, a contradiction.