Why for every rational $p$ the level sets $\{ f \leq p \}$ and $\{ f \geq p \}$ are $\prod_{\xi+1}^0$ sets?

Question

$f$ is a real-valued baire class $\xi$ function if $\{ f < c \}$ and $\{ f >c \}$ are in $\sum_{\xi +1}^0$ for every $c \in \mathbb{R}$.

In the proof of Proposition $5.2$ page $24$, we have the following sentence:

For every rational $p$ the level sets $\{ f \leq p \}$ and $\{ f \geq p \}$ are $\prod_{\xi+1}^0$ sets.

Why the statement above true? I think it follows by the definition of baire class $\xi$ function? Am I right?

Answer 1

You're over-thinking this: the complement of a $\bf \Sigma^0_\theta$ set is $\bf \Pi^0_\theta$, and $\{f\le p\}=\mathbb{R}\setminus \{f>p\}$ etc.