Exercise 3.2.15

Question

A set $A$ is called an $F_{\sigma}$ set if it can be written as the countable union of closed sets. A set $B$ is called a $G_{\delta}$ set if it can be written as the countable intersection of open sets.
  \begin{enumerate}[label=(\alph*)] 
  \item Show that a closed interval $[a, b]$ is a $G_{\delta}$ set.
  \item Show that the half-open interval $(a, b]$ is both a $G_{\delta}$ and an $F_{\sigma}$ set.
  \item Show that $\mathbf{Q}$ is an $F_{\sigma}$ set, and the set of irrationals $\mathbf{I}$ forms a $G_{\delta}$ set. (We will see in Section $3.5$ that $\mathbf{Q}$ is not a $G_{\delta}$ set, nor is $\mathbf{I}$ an $F_{\sigma}$ set.)
   \end{enumerate}

Ulisse Mini · Accepted Answer

\begin{enumerate}[label=(\alph*)] 
  \item $[a,b] = \bigcap_{n=1}^\infty (a-1/n, b+1/n)$
  \item $(a,b] = \bigcap_{n=1}^\infty (a, b+1/n) = \bigcup_{n=1}^\infty [a+1/n, b]$
  \item Let $r_n$ be an enumeration of $\mathbf Q$ (possible since $\mathbf Q$ is countable), we have
    $$\mathbf Q = \bigcup_{n=1}^\infty [r_n, r_n]$$
    Applying De Morgan's laws combined with the complement of a closed set being open we get
    $$\mathbf Q^c = \bigcap_{n=1}^\infty [r_n, r_n]^c$$
   \end{enumerate}

Exercise 3.2.15

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Exercise 3.2.15

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