Exercise 3.5.2

Question

Replace each with the word finite or countable, depending on which is more appropriate.
  \begin{enumerate}[label=(\alph*)] 
  \item The \makebox[1cm]{\hrulefill}\space union of $F_{\sigma}$ sets is an $F_{\sigma}$ set.
  \item The \makebox[1cm]{\hrulefill}\space intersection of $F_{\sigma}$ sets is an $F_{\sigma}$ set.
  \item The \makebox[1cm]{\hrulefill}\space union of $G_{\delta}$ sets is a $G_{\delta}$ set.
  \item The \makebox[1cm]{\hrulefill}\space intersection of $G_{\delta}$ sets is a $G_{\delta}$ set.
   \end{enumerate}

Ulisse Mini · Accepted Answer

\begin{enumerate}[label=(\alph*)] 
  \item Countable, since two countable union can be written as a single countable union over the diagonal (see Exercise 1.2.4). Another way of seeing this is that we can form a bijection between $\mathbf N$ and $\mathbf N^2$, therefore a double infinite union can be written as a single infinite union.
  \item Finite
  \item Finite
  \item Countable, by the same logic as in (a) we can write two countable intersections as a single countable intersection.
   \end{enumerate}

Exercise 3.5.2

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Exercise 3.5.2

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